| 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 |
| 49.3-a1 |
49.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{8} \) |
$1.27317$ |
$(a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$1$ |
$10.04629340$ |
1.865549852 |
\( -12288 a + 40960 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 10 a - 31\) , \( 40 a - 132\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10a-31\right){x}+40a-132$ |
| 49.3-b1 |
49.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1.575368639$ |
$1.459284802$ |
1.707588603 |
\( -1407628760845 a - 3086342051803 \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -122 a - 265\) , \( -1292 a - 2825\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-122a-265\right){x}-1292a-2825$ |
| 49.3-b2 |
49.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.525122879$ |
$4.377854408$ |
1.707588603 |
\( -3515 a - 7688 \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -2 a\) , \( -3 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}-2a{x}-3a-5$ |
| 49.3-b3 |
49.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.315073727$ |
$7.296424013$ |
1.707588603 |
\( 1407628760845 a - 4493970812648 \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -108 a - 236\) , \( -9766 a - 21413\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-108a-236\right){x}-9766a-21413$ |
| 49.3-b4 |
49.3-b |
$4$ |
$15$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.105024575$ |
$21.88927204$ |
1.707588603 |
\( 3515 a - 11203 \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 3 a - 8\) , \( -8 a + 26\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-8\right){x}-8a+26$ |
| 49.3-c1 |
49.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{2} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$0.107949430$ |
$47.01921204$ |
1.885066603 |
\( -12288 a + 40960 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( a - 3\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a-3\right){x}-2a+2$ |
| 49.3-d1 |
49.3-d |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{8} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$5.989773894$ |
$0.782570221$ |
1.740863594 |
\( -\frac{213433415640625}{49} a + \frac{97343395248346}{7} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a - 230\) , \( 259 a - 1669\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a-230\right){x}+259a-1669$ |
| 49.3-d2 |
49.3-d |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{16} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1.197954778$ |
$3.912851106$ |
1.740863594 |
\( -\frac{91176666325}{282475249} a + \frac{41583934921}{40353607} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 40 a - 124\) , \( -23 a + 86\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(40a-124\right){x}-23a+86$ |
| 49.3-d3 |
49.3-d |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{11} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$0.598977389$ |
$7.825702212$ |
1.740863594 |
\( \frac{173650213}{16807} a + \frac{58516320}{2401} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -10 a + 31\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-10a+31\right){x}$ |
| 49.3-d4 |
49.3-d |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{7} \) |
$1.27317$ |
$(a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.994886947$ |
$1.565140442$ |
1.740863594 |
\( \frac{94831363}{7} a + 24861195 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -28 a - 75\) , \( -138 a - 355\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-28a-75\right){x}-138a-355$ |
*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.
