| 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 |
| 507.1-a1 |
507.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.609052866$ |
3.815738451 |
\( -\frac{466432}{1521} a + \frac{730048}{1521} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -21 a + 35\) , \( 42 a - 73\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-21a+35\right){x}+42a-73$ |
| 507.1-a2 |
507.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$13.21810573$ |
3.815738451 |
\( \frac{30751232}{507} a + \frac{54701888}{507} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 2 a - 2\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(2a-2\right){x}-1$ |
| 507.1-b1 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( - 3 \cdot 13^{10} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.467302419$ |
1.578274261 |
\( -\frac{55138519571522419}{2447192163} a + \frac{31834375272105960}{815730721} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 95 a - 139\) , \( -507 a + 1036\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(95a-139\right){x}-507a+1036$ |
| 507.1-b2 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$10.93460483$ |
1.578274261 |
\( \frac{12167}{39} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -27 a + 46\) , \( 230 a - 399\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-27a+46\right){x}+230a-399$ |
| 507.1-b3 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{4} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.93460483$ |
1.578274261 |
\( \frac{10218313}{1521} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -4\) , \( -5\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-4{x}-5$ |
| 507.1-b4 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{8} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$10.93460483$ |
1.578274261 |
\( \frac{822656953}{85683} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -19\) , \( 22\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-19{x}+22$ |
| 507.1-b5 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{8} \cdot 13^{2} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.733651209$ |
1.578274261 |
\( \frac{37159393753}{1053} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -69\) , \( -252\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-69{x}-252$ |
| 507.1-b6 |
507.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( - 3 \cdot 13^{10} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.467302419$ |
1.578274261 |
\( \frac{55138519571522419}{2447192163} a + \frac{31834375272105960}{815730721} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -95 a - 139\) , \( 507 a + 1036\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-95a-139\right){x}+507a+1036$ |
| 507.1-c1 |
507.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.609052866$ |
3.815738451 |
\( \frac{466432}{1521} a + \frac{730048}{1521} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -2\) , \( 3 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}-2{x}+3a-6$ |
| 507.1-c2 |
507.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$13.21810573$ |
3.815738451 |
\( -\frac{30751232}{507} a + \frac{54701888}{507} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -a - 3\) , \( -3 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-a-3\right){x}-3a-6$ |
| 507.1-d1 |
507.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.111435177$ |
$15.59701041$ |
1.003466878 |
\( \frac{466432}{1521} a + \frac{730048}{1521} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( a - 1\) , \( -2 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(a-1\right){x}-2a+4$ |
| 507.1-d2 |
507.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.055717588$ |
$31.19402083$ |
1.003466878 |
\( -\frac{30751232}{507} a + \frac{54701888}{507} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -1\) , \( -a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}-{x}-a-1$ |
| 507.1-e1 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( - 3 \cdot 13^{10} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.565167813$ |
$1.307121894$ |
1.706054394 |
\( -\frac{55138519571522419}{2447192163} a + \frac{31834375272105960}{815730721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 95 a - 140\) , \( 602 a - 1176\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(95a-140\right){x}+602a-1176$ |
| 507.1-e2 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.130335626$ |
$20.91395031$ |
1.706054394 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -27 a + 47\) , \( -230 a + 398\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-27a+47\right){x}-230a+398$ |
| 507.1-e3 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{4} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.565167813$ |
$20.91395031$ |
1.706054394 |
\( \frac{10218313}{1521} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}$ |
| 507.1-e4 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{8} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.282583906$ |
$5.228487579$ |
1.706054394 |
\( \frac{822656953}{85683} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -20\) , \( -42\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-20{x}-42$ |
| 507.1-e5 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{8} \cdot 13^{2} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.282583906$ |
$20.91395031$ |
1.706054394 |
\( \frac{37159393753}{1053} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -70\) , \( 182\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-70{x}+182$ |
| 507.1-e6 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( - 3 \cdot 13^{10} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.565167813$ |
$1.307121894$ |
1.706054394 |
\( \frac{55138519571522419}{2447192163} a + \frac{31834375272105960}{815730721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -95 a - 140\) , \( -602 a - 1176\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-95a-140\right){x}-602a-1176$ |
| 507.1-f1 |
507.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{4} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.111435177$ |
$15.59701041$ |
1.003466878 |
\( -\frac{466432}{1521} a + \frac{730048}{1521} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -20 a + 36\) , \( -62 a + 108\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-20a+36\right){x}-62a+108$ |
| 507.1-f2 |
507.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{3} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.055717588$ |
$31.19402083$ |
1.003466878 |
\( \frac{30751232}{507} a + \frac{54701888}{507} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 3 a - 2\) , \( -a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-2\right){x}-a+3$ |
*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.
