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 |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{12} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( -\frac{132651}{2} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -11 a - 19\) , \( 26 a + 44\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11a-19\right){x}+26a+44$ |
324.1-a2 |
324.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{4} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( -\frac{1167051}{512} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -5 a - 9\) , \( -13 a - 23\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-5a-9\right){x}-13a-23$ |
324.1-a3 |
324.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{12} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( \frac{9261}{8} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -4 a + 11\) , \( 5 a - 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-4a+11\right){x}+5a-13$ |
324.1-b1 |
324.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$39.86878607$ |
2.900027461 |
\( -\frac{132651}{2} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-3{x}+3$ |
324.1-b2 |
324.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.476621706$ |
2.900027461 |
\( -\frac{1167051}{512} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 67 a - 189\) , \( 702 a - 1960\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(67a-189\right){x}+702a-1960$ |
324.1-b3 |
324.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$13.28959535$ |
2.900027461 |
\( \frac{9261}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -8 a + 21\) , \( -18 a + 50\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a+21\right){x}-18a+50$ |
324.1-c1 |
324.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{12} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( -\frac{132651}{2} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 9 a - 30\) , \( -27 a + 70\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(9a-30\right){x}-27a+70$ |
324.1-c2 |
324.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{4} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( -\frac{1167051}{512} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 4 a - 14\) , \( 13 a - 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-14\right){x}+13a-36$ |
324.1-c3 |
324.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{12} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$6.506893680$ |
1.419920610 |
\( \frac{9261}{8} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 2 a + 8\) , \( -6 a - 8\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a+8\right){x}-6a-8$ |
324.1-d1 |
324.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$3.185925848$ |
0.695226017 |
\( -\frac{132651}{2} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 15 a - 42\) , \( 72 a - 201\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-42\right){x}+72a-201$ |
324.1-d2 |
324.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$9.557777544$ |
0.695226017 |
\( -\frac{1167051}{512} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -14\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-14{x}+29$ |
324.1-d3 |
324.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.73734$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$9.557777544$ |
0.695226017 |
\( \frac{9261}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 1\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+{x}-1$ |
*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.