Distribution
of analytic square root of Sha values on negative quadratic
twists of elliptic curves
These are graphs for the distribution of analytic square root values of
Sha in families of negative quadratic twists of elliptic curves.
E=[ a1,a2,a3,a4,a6]
denotes an elliptic curve of prime conductor Nand rank zero. For
d>0, a
fundamental discriminant, let Ed denote the -d quadratic
twists of E and |Shad| the order of its Tate-Shafarevich
group. Then there is a weight 3/2 modular form g related to E such that
the d-coefficient of its Fourier expansion, md, is
related to the value at s=1 of the L-series of Ed. Assuming
the Birch and Swinnerton-Dyer conjecture permits to calculate the order
of Shad. Specifically there is a relationship |Shad|
= md2 qd2. By square root
of |Shad| we mean the (integer) number md qd.
The graphs for the distribution of these values are denoted by
TwistNrtshadist.ps where N is the conductor.
The weight 3/2 modular form g is a linear combination of theta forms
denoted by gi.
GENERAL: ordered by conductor the
information is as follows:
N is de conductor, E is the elliptic curve of conductor N considered,
|Tors(E)| the order of the group of torsion points, tot is the range of
d's considered 0 < d < tot
TwistNrtshadist.ps is the distribution of square roots of sha for the
elliptic curve E of conductor N: [n, #{d such that sqrt(sha_d)=n}]
TwistNrtshaprm.ps is the same as above but removing from the list those
sha values divided by p ("prm" is "p removed") when E has a point of
odd order p defined over Q.
TwistNrtshalog.ps is the graph for the logarithm of the distribution:
[n,log(#{d such that sqrt(sha_d)=n}]
TwistNoddrtshalog.ps the same as above for n odd and the best linear
approximation of the graph.
TwistNevenrtshalog.ps same as above, now for n even.
TwistNdistgi.ps is the distribution of the coefficients of the theta
forms g_i involved in the linear combination that gives g. Ai is the
matrix of the corresponding ternary form and #Aut the number of its
automorphisms.
TwistNdistg.ps is the graph for the distribution of the coefficients of
the weight 3/2 modular form g.
TwistNdistG.ps is the distribution of the coefficients of W*G, where G
is the weight 3/2 Eisenstein series and W one half the product of the
number of units in each order R_i.
N=11
E=[0,-1,1,-10,-20] |Tors(E)|=5 tot=10 million
Twist11rtshadist.ps
distribution of square root of Sha values of imaginary quadratic
twists of elliptic curve E.
Twist11rtshadistop.ps same as above,
only positive odd values of sha.
Twist11rtsha5rm.ps
same as above, but with Sha values divided by 5=|Tors(E)| removed.
Twist11rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist11oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist11evenrtshalog.ps
same as above but for even values
Twist11oddrtshalogapp.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by a curve of type:
u - 1/s x - c log(1+x); [u=0.146; s=23.870; c=0.687].
Twist11evenrtshalogapp.ps
same as above but for even values [u=0.050; s=10.803; c=0.209].
Twist11distg1.ps
distribution of coefficients of theta series given by ternary form
[11,0,0;0,12,22;0,22,44] (4 automorphisms)
Twist11distg2.ps
same as above, now for ternary form [3,1,1;1,15,-7;1,-7,15]
(6 automorphisms)
Twist11distg.ps
distribution of coefficients of weight 3/2 modular form
g=-g1+g2 associated to E
Twist11distG.ps
distribution of coefficients of weight 3/2 Eisenstein series 6G=3g1+2g2.
N=17 E=[1,-1,1,-1,-14] |Tors(E)|=4 tot=10 million
Twist17rtshadist.ps distribution of
square root of Sha values of
imaginary quadratic twists of elliptic curve E.
Twist17rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist17oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist17evenrtshalog.ps
same as above but for even values
Twist17oddrtshalogapp.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by a curve of type:
u - 1/s x - c log(1+x); [u=0.177; s=16.118; c=0.900].
Twist17evenrtshalogapp.ps
same as above but for even values [u=0.094; s=6.338; c=0.042].
Twist17distg1.ps
distribution of coefficients of theta series given by ternary form
[3,-1,1;-1,23,11;1,11,23] (6 aut)
Twist17distg2.ps
same as above, now for ternary form [7,-3,-2;-3,11,-4;-2,-4,20]
(4 aut)
Twist17distg.ps
distribution of coefficients of weight 3/2 modular form g=-g1+g2
associated to E.
Twist17distG.ps
distribution of coefficients of weight 3/2 Eisenstein series 3G=g1+3g2.
N=19 E=[0,1,1,-9,-15] |Tors(E)|=3 tot=3 million
Twist19rtshadist.ps
distribution of square root of Sha values of
imaginary quadratic twists of elliptic curve E.
Twist19rtsha3rm.ps
same as above, but with Sha values divided by 3=|Tors(E)| removed.
Twist19rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist19oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist19evenrtshalog.ps
same as above but for even values
Twist19distg1.ps
distribution of coefficients of theta series given by ternary form
[4,2,0;2,20,0;0,0,19] (4 aut)
Twist19distg2.ps
same as above, now for ternary form [7,-1,-3;-1,11,-5;-3,-5,23]
(2 aut)
Twist19distg.ps
distribution of coefficients of weight 3/2 modular form
g=-g1+g2 associated to E.
Twist19distG.ps
distribution of coefficients of weight 3/2 Eisenstein series 2G=g1+2g2.
N=37 E=[0,1,1,-23,-50] |Tors(E)|=3 tot=10 million
Twist37rtshadist.ps
Distribution of square root of Sha values of
imaginary quadratic twists of elliptic curve E.
Twist37rtsha3rm.ps
same as above, but with Sha values divided by 3=|Tors(E)| removed.
Twist37rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist37oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist37evenrtshalog.ps
same as above but for even values
Twist37distg1.ps
distribution of coefficients of theta series given by ternary form
[8,2,-4;2,19,-1;-4,-1,39] (2 aut)
Twist37distg2.ps
same as above, now for ternary form [15,7,-2;7,23,4;-2,4,20]
(1 aut)
Twist37distg.ps
distribution of coefficients of weight 3/2 modular form g=-2g1+2g2
associated to E
N=67
E=[0,1,1,-12,-21] |Tors(E)|=1 tot=10 million
Twist67rtshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve E.
Twist67rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist67oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist67evenrtshalog.ps
same as above but for even values
Twist67distg1.ps
distribution of coefficients of theta series given by ternary form
[4,2,0;2,68,0;0,0,67] (4 aut)
Twist67distg2.ps distribution of
coefficients of theta series given by ternary form
[23,-4,1;-4,24,-6;1,-6,35] (1 aut)
Twist67distg4.ps
same as above, now for ternary form [16,6,-8;6,19,-3;-8,-3,71] (2 aut)
Twist67distg5.ps
same as above, now for ternary form [15,2,7;2,36,-8;7,-8,39] (1 aut)
Twist67distg.ps
distribution of coefficients of weight 3/2 modular form g=-g1-g4+2g5
associated to E
Twist67distG.ps
distribution of coefficients of weight 3/2 Eisenstein series
2G=g1+4g2+2g4+4g5
N=73 E=[1,-1,0,4,-3] |Tors(E)|=2 tot=3 million
Twist73rtshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve E.
Twist73rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist73oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist73evenrtshalog.ps
same as above but for even values
Twist73distg1.ps
distribution of coefficients of theta series given by ternary form
[7,3,-2;3,43,20;-2,20,84] (2 aut)
Twist73distg2.ps
same as above, now for ternary form [15,4,1;4,40,10;1,10,39]
(1 aut)
Twist73distg3.ps same as above, now for ternary form
[20,6,-8;6,31,-17;-8,-17,47] (1 aut)
Twist73distg5.ps
same as above, now for ternary form [11,-4,-2;-4,28,14;-2,14,80] (2 aut)
Twist73distg.ps
distribution of coefficients of weight 3/2 modular form g=g1+2g2-2g3-g5
associated to E
N=89 E=[1,1,0,4,5] |Tors(E)|=2 tot=3 million
Twist89rtshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve E.
Twist89rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist89oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist89evenrtshalog.ps
same as above but for even values
Twist89distg1.ps
distribution of coefficients of theta series given by ternary form
[3,-1,1;-1,119,59;1,59,119] (6 aut)
Twist89distg2.ps
same as above, now for ternary form [12,4,6;4,31,2;6,2,92]
(2 aut)
Twist89distg3.ps same as above, now for ternary form
[15,7,4;7,27,-10;4,-10,96] (2
aut)
Twist89distg4.p
s same as above, now for ternary form [7,-1,-3;-1,51,-25;-3,-25,103] (2
aut)
Twist89distg5.ps
same as above, now for ternary form [15,-2,1;-2,24,-12;1,-12,95]
(2 aut)
Twist89distg6.ps same as above, now for ternary form
[23,1,-6;1,31,-8;-6,-8,48]
(1 aut)
Twist89distg7.ps same as above, now for ternary form
[19,9,5;9,23,-7;5,-7,95] (2 aut)
N=109
E=[1,-1,0,-8,-7] |Tors(E)|=1 tot=3 million
Twist109rtshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve E.
Twist109rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist109oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist109evenrtshalog.ps
same as above but for even values
Twist109distg1.ps
distribution of coefficients of theta series given by ternary form
[8,2,-4;2,55,-1;-4,-1,111] (2 aut)
Twist109distg2.ps
same as above, now for ternary form [32,-10,-14;-10,44,18,-14,18,47]
(1 aut)
Twist109distg3.ps same as above, now
for ternary form
[11,2,1;2,40,20;1,20,119] (2
aut)
Twist109distg5.ps
same as above, now for ternary form [23,5,-7;5,39,-11;-7,-11,59]
(1 aut)
Twist109distg6.ps same as above, now
for ternary form
[19,1,-9;1,23,11;-9,11,119]
(2 aut)
Twist109distg7.ps same as above, now
for ternary form
[24,8,6;8,39,2;6,2,56] (1 aut)
Twist109distg.ps
distribution of coefficients of weight 3/2 modular form g=-g3-g6-+2g7
associated to E
N=139 E=[1,1,0,-3,-4] |Tors(E)|=1 tot=3 million
Twist139rtshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve E.
Twist139rtshalog.ps
logarithm of distribution (only positive values of sqrt(Sha))
Twist139oddrtshalog.ps
logarithm of distribution for odd values of sqrt(Sha) and best
approximation by straight line
Twist139evenrtshalog.ps
same as above but for even values
N=307 There are four
isogeny classes, each with an elliptic curve of rank zero. As in
Cremona's tables we list them 307A - 307D.
307A=[0,0,1,-8,-9] |Tors(E)|=1 tot=3 million
307B=[1,1,0,0,-1] |Tors(E)|=1 tot=3 million
307C=[0,0,1,1,-1] |Tors(E)|=1 tot=3 million
307D=[0,-1,1,2,-1] |Tors(E)|=1 tot=3 million
Twist307Artshadist.ps
Distribution of square root of Sha values of imaginary quadratic twists
of elliptic curve 307A.
Twist307Brtshadist.ps Same as above
for elliptic curve 307B.
Twist307Crtshadist.ps Same as above
for elliptic curve 307C.
Twist307Drtshadist.ps Same as above
for elliptic curve 307D.
Twist307oddrtshaABCD.ps Superposition of
four graphs above for positive odd rtsha values.
Twist307Aoddrtshalog.ps
log of distribution for odd values of sqrt(Sha) and best
approximation by straight line for curve 307A.
Twist307Boddrtshalog.ps Same as
above for curve 307B.
Twist307Coddrtshalog.ps Same as
above for for curve 307C.
Twist307Doddrtshalog.ps Same as
above for for curve 307D.
Twist307Aevenrtshalog.ps
log of distribution for even values of sqrt(Sha) and best
approximation by straight line for curve 307A.
Twist307Bevenrtshalog.ps
same as above for curve 307B.
Twist307Cevenrtshalog.ps
same as above for curve 307C.
Twist307Devenrtshalog.ps
same as above for curve 307D.
Other
graphs
Twist11-89rtshadist.ps Superposition
of graphs TwistNrtshadist for N=11,17,19,37,67,73,89.