| 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 |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{34} \) |
$0.92001$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$8.120880803$ |
1.115488766 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -40 a - 124\) , \( 801 a + 2515\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-40a-124\right){x}+801a+2515$ |
| 196.2-d1 |
196.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
196.2 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{34} \cdot 7^{6} \) |
$2.43411$ |
$(-a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$4.857662064$ |
$1.473724687$ |
3.933378267 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 52 a - 230\) , \( -1242 a + 5122\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(52a-230\right){x}-1242a+5122$ |
| 196.3-d1 |
196.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
196.3 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{34} \cdot 7^{6} \) |
$2.43411$ |
$(-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$4.857662064$ |
$1.473724687$ |
3.933378267 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -54 a - 177\) , \( 1241 a + 3881\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-54a-177\right){x}+1241a+3881$ |
| 256.1-g1 |
256.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{58} \) |
$2.60217$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$3.292102894$ |
$0.974777255$ |
3.526394050 |
\( -\frac{25153757}{131072} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 93\) , \( 291 a - 192\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-93\right){x}+291a-192$ |
| 256.1-o1 |
256.1-o |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{58} \) |
$2.60217$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$3.292102894$ |
$0.974777255$ |
3.526394050 |
\( -\frac{25153757}{131072} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a - 93\) , \( -291 a + 192\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-93\right){x}-291a+192$ |
| 256.1-v1 |
256.1-v |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{58} \) |
$2.60217$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.468023467$ |
0.128575934 |
\( -\frac{25153757}{131072} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -684 a - 2144\) , \( -59468 a - 186736\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-684a-2144\right){x}-59468a-186736$ |
| 324.1-d1 |
324.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{34} \cdot 3^{12} \) |
$2.76002$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 2^{2} \) |
$1$ |
$0.624031290$ |
3.085822437 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -386 a - 1211\) , \( -24677 a - 77489\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-386a-1211\right){x}-24677a-77489$ |
| 484.2-b1 |
484.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{34} \cdot 11^{6} \) |
$3.05132$ |
$(a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 17 \) |
$1$ |
$1.175625604$ |
2.745238131 |
\( -\frac{25153757}{131072} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 195 a - 808\) , \( 9219 a - 38165\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(195a-808\right){x}+9219a-38165$ |
| 484.3-b1 |
484.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{34} \cdot 11^{6} \) |
$3.05132$ |
$(a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 17 \) |
$1$ |
$1.175625604$ |
2.745238131 |
\( -\frac{25153757}{131072} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -196 a - 612\) , \( -9220 a - 28945\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-196a-612\right){x}-9220a-28945$ |
| 676.2-d1 |
676.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{34} \cdot 13^{6} \) |
$3.31714$ |
$(a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 17 \) |
$0.278911152$ |
$2.252327087$ |
5.867705636 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -8 a - 84\) , \( 102 a + 823\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8a-84\right){x}+102a+823$ |
| 676.3-d1 |
676.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{34} \cdot 13^{6} \) |
$3.31714$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 17 \) |
$0.278911152$ |
$2.252327087$ |
5.867705636 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 6 a - 90\) , \( -103 a + 926\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(6a-90\right){x}-103a+926$ |
| 1156.2-c1 |
1156.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.2 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{34} \cdot 17^{6} \) |
$3.79329$ |
$(a+5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 1 \) |
$1$ |
$1.969602901$ |
2.434911887 |
\( -\frac{25153757}{131072} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 146 a - 617\) , \( -5890 a + 24416\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(146a-617\right){x}-5890a+24416$ |
| 1156.3-c1 |
1156.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1156.3 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{34} \cdot 17^{6} \) |
$3.79329$ |
$(a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$9$ |
\( 1 \) |
$1$ |
$1.969602901$ |
2.434911887 |
\( -\frac{25153757}{131072} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -147 a - 470\) , \( 5889 a + 18527\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-147a-470\right){x}+5889a+18527$ |
*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.
