Label Class Class size Class degree Base field Field degree Field signature Conductor Conductor norm Discriminant norm Root analytic conductor Bad primes Rank Torsion CM CM Sato-Tate $\Q$-curve Base change Semistable Potentially good Nonmax $\ell$ mod-$\ell$ images $Ш_{\textrm{an}}$ Tamagawa Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
19.3-a1 19.3-a $1$
$1$
4.4.15125.1 $4$
$[4, 0]$
19.3
\( 19 \)
\( - 19^{3} \)
$15.87927$
$(6a^3+11a^2-53a-65)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 3 \)
$0.019605404$
$1650.734053$
3.157813394
\( \frac{39656807629}{6859} a^{3} + \frac{69710284663}{6859} a^{2} - \frac{362938149300}{6859} a - \frac{445724222643}{6859} \)
\( \bigl[a^{3} + 2 a^{2} - 8 a - 13\) , \( 4 a^{3} + 7 a^{2} - 37 a - 43\) , \( -2 a^{3} - 3 a^{2} + 18 a + 17\) , \( -6 a^{3} + a^{2} + 39 a + 27\) , \( -a^{3} + 41 a^{2} - 62 a - 132\bigr] \) ${y}^2+\left(a^{3}+2a^{2}-8a-13\right){x}{y}+\left(-2a^{3}-3a^{2}+18a+17\right){y}={x}^{3}+\left(4a^{3}+7a^{2}-37a-43\right){x}^{2}+\left(-6a^{3}+a^{2}+39a+27\right){x}-a^{3}+41a^{2}-62a-132$
19.3-b1 19.3-b $1$
$1$
4.4.15125.1 $4$
$[4, 0]$
19.3
\( 19 \)
\( - 19^{3} \)
$15.87927$
$(6a^3+11a^2-53a-65)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 1 \)
$1$
$242.4834118$
1.971670517
\( \frac{39656807629}{6859} a^{3} + \frac{69710284663}{6859} a^{2} - \frac{362938149300}{6859} a - \frac{445724222643}{6859} \)
\( \bigl[a^{3} + 2 a^{2} - 8 a - 12\) , \( -a - 1\) , \( -2 a^{3} - 3 a^{2} + 19 a + 17\) , \( 5 a^{3} + 10 a^{2} - 47 a - 64\) , \( 4 a^{3} + 9 a^{2} - 38 a - 57\bigr] \) ${y}^2+\left(a^{3}+2a^{2}-8a-12\right){x}{y}+\left(-2a^{3}-3a^{2}+19a+17\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a^{3}+10a^{2}-47a-64\right){x}+4a^{3}+9a^{2}-38a-57$
19.3-c1 19.3-c $1$
$1$
4.4.15125.1 $4$
$[4, 0]$
19.3
\( 19 \)
\( - 19^{3} \)
$15.87927$
$(6a^3+11a^2-53a-65)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 1 \)
$1$
$58.43384051$
0.475134689
\( \frac{39656807629}{6859} a^{3} + \frac{69710284663}{6859} a^{2} - \frac{362938149300}{6859} a - \frac{445724222643}{6859} \)
\( \bigl[a^{3} + 2 a^{2} - 9 a - 12\) , \( 2 a^{3} + 3 a^{2} - 19 a - 18\) , \( -2 a^{3} - 3 a^{2} + 19 a + 17\) , \( -7 a^{3} - 11 a^{2} + 64 a + 65\) , \( 3 a^{3} + 11 a^{2} - 24 a - 91\bigr] \) ${y}^2+\left(a^{3}+2a^{2}-9a-12\right){x}{y}+\left(-2a^{3}-3a^{2}+19a+17\right){y}={x}^{3}+\left(2a^{3}+3a^{2}-19a-18\right){x}^{2}+\left(-7a^{3}-11a^{2}+64a+65\right){x}+3a^{3}+11a^{2}-24a-91$
19.3-d1 19.3-d $1$
$1$
4.4.15125.1 $4$
$[4, 0]$
19.3
\( 19 \)
\( - 19^{3} \)
$15.87927$
$(6a^3+11a^2-53a-65)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 3 \)
$0.097238852$
$397.7951713$
3.774269320
\( \frac{39656807629}{6859} a^{3} + \frac{69710284663}{6859} a^{2} - \frac{362938149300}{6859} a - \frac{445724222643}{6859} \)
\( \bigl[-3 a^{3} - 5 a^{2} + 27 a + 31\) , \( 4 a^{3} + 7 a^{2} - 37 a - 43\) , \( a^{3} + 2 a^{2} - 8 a - 13\) , \( -2 a^{3} + 11 a^{2} + 23 a - 105\) , \( 6 a^{3} - 16 a^{2} - 73 a + 196\bigr] \) ${y}^2+\left(-3a^{3}-5a^{2}+27a+31\right){x}{y}+\left(a^{3}+2a^{2}-8a-13\right){y}={x}^{3}+\left(4a^{3}+7a^{2}-37a-43\right){x}^{2}+\left(-2a^{3}+11a^{2}+23a-105\right){x}+6a^{3}-16a^{2}-73a+196$
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Pari/GP
SageMath
Magma
Oscar
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*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
