| 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 |
| 324.1-a1 |
324.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{6} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$5.559532991$ |
$5.898343969$ |
4.943585718 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -1\bigr] \) |
${y}^2={x}^{3}-1$ |
| 324.1-a2 |
324.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{18} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.853177663$ |
$5.898343969$ |
4.943585718 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 27\bigr] \) |
${y}^2={x}^{3}+27$ |
| 324.1-a3 |
324.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{6} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$2.779766495$ |
$11.79668793$ |
4.943585718 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -222 a - 735\) , \( -3887 a - 12892\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-222a-735\right){x}-3887a-12892$ |
| 324.1-a4 |
324.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.926588831$ |
$11.79668793$ |
4.943585718 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 2027 a - 6711\) , \( -90208 a + 299200\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(2027a-6711\right){x}-90208a+299200$ |
| 324.1-b1 |
324.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.747009330$ |
$16.40727385$ |
3.695439622 |
\( -9344 a + \frac{115792}{3} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -7 a - 27\) , \( 2 a + 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-7a-27\right){x}+2a+8$ |
| 324.1-b2 |
324.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.494018661$ |
$8.203636928$ |
3.695439622 |
\( \frac{21118868}{3} a + \frac{210149176}{9} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -142 a - 477\) , \( 1343 a + 4454\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-142a-477\right){x}+1343a+4454$ |
| 324.1-c1 |
324.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.494018661$ |
$8.203636928$ |
3.695439622 |
\( -\frac{21118868}{3} a + \frac{210149176}{9} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 147 a - 471\) , \( -1820 a + 6041\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(147a-471\right){x}-1820a+6041$ |
| 324.1-c2 |
324.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.747009330$ |
$16.40727385$ |
3.695439622 |
\( 9344 a + \frac{115792}{3} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 12 a - 21\) , \( -29 a + 110\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(12a-21\right){x}-29a+110$ |
| 324.1-d1 |
324.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.329787359$ |
$6.585481807$ |
1.964474842 |
\( -9344 a + \frac{115792}{3} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -6 a - 21\) , \( -47 a - 156\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-6a-21\right){x}-47a-156$ |
| 324.1-d2 |
324.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.659574718$ |
$3.292740903$ |
1.964474842 |
\( \frac{21118868}{3} a + \frac{210149176}{9} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -141 a - 471\) , \( -2108 a - 6987\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-141a-471\right){x}-2108a-6987$ |
| 324.1-e1 |
324.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{6} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.247247529$ |
$17.69503190$ |
2.887481112 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 1\bigr] \) |
${y}^2={x}^{3}+1$ |
| 324.1-e2 |
324.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{18} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$9.741742587$ |
$1.966114656$ |
2.887481112 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -27\bigr] \) |
${y}^2={x}^{3}-27$ |
| 324.1-e3 |
324.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{6} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.623623764$ |
$35.39006381$ |
2.887481112 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -223 a - 741\) , \( 2696 a + 8940\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-223a-741\right){x}+2696a+8940$ |
| 324.1-e4 |
324.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$4.870871293$ |
$3.932229312$ |
2.887481112 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 2028 a - 6705\) , \( 87547 a - 290342\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(2028a-6705\right){x}+87547a-290342$ |
| 324.1-f1 |
324.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{16} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.659574718$ |
$3.292740903$ |
1.964474842 |
\( -\frac{21118868}{3} a + \frac{210149176}{9} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 146 a - 477\) , \( 1631 a - 5406\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(146a-477\right){x}+1631a-5406$ |
| 324.1-f2 |
324.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51479$ |
$(a+3), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.329787359$ |
$6.585481807$ |
1.964474842 |
\( 9344 a + \frac{115792}{3} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 11 a - 27\) , \( 20 a - 60\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11a-27\right){x}+20a-60$ |
*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.
