| 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 |
| 21609.3-CMd1 |
21609.3-CMd |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{20} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
✓ |
|
✓ |
$7$ |
7Cs.6.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.316556078$ |
1.462109897 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -4202\bigr] \) |
${y}^2+{y}={x}^{3}-4202$ |
| 21609.3-CMc1 |
21609.3-CMc |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{8} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$7$ |
7Cs.6.1 |
$1$ |
\( 1 \) |
$1$ |
$2.215892550$ |
2.558692321 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -6 a + 14\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-6a+14$ |
| 21609.3-CMb1 |
21609.3-CMb |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{8} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
✓ |
|
✓ |
$3, 7$ |
3B.1.1[2], 7Cs.6.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.066644097$ |
$2.215892550$ |
2.728347852 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 12\bigr] \) |
${y}^2+{y}={x}^{3}+12$ |
| 21609.3-CMa1 |
21609.3-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{8} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-3$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$7$ |
7Cs.6.1 |
$1$ |
\( 1 \) |
$1$ |
$2.215892550$ |
2.558692321 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 5 a + 9\bigr] \) |
${y}^2+a{y}={x}^{3}+5a+9$ |
| 21609.3-a1 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{19} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{5} \) |
$1$ |
$0.139791446$ |
1.291338071 |
\( \frac{308817493407}{2401} a - \frac{61895989485}{2401} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -14122 a - 799\) , \( -780754 a + 343883\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-14122a-799\right){x}-780754a+343883$ |
| 21609.3-a2 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{25} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{5} \) |
$1$ |
$0.139791446$ |
1.291338071 |
\( \frac{2097781165791}{13841287201} a - \frac{295085536866}{13841287201} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -672 a - 579\) , \( 52809 a - 36737\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-672a-579\right){x}+52809a-36737$ |
| 21609.3-a3 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{25} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{5} \) |
$1$ |
$0.139791446$ |
1.291338071 |
\( -\frac{2097781165791}{13841287201} a + \frac{1802695628925}{13841287201} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 578 a + 671\) , \( -52810 a + 16073\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(578a+671\right){x}-52810a+16073$ |
| 21609.3-a4 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{16} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.559165787$ |
1.291338071 |
\( \frac{988929}{343} a + \frac{1141344}{343} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 63 a + 156\) , \( -846 a + 895\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(63a+156\right){x}-846a+895$ |
| 21609.3-a5 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{20} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B[2] |
$1$ |
\( 2^{6} \) |
$1$ |
$0.279582893$ |
1.291338071 |
\( \frac{854150427}{117649} a + \frac{702560952}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -892 a - 64\) , \( -11944 a + 5636\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-892a-64\right){x}-11944a+5636$ |
| 21609.3-a6 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{16} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.559165787$ |
1.291338071 |
\( -\frac{988929}{343} a + \frac{2130273}{343} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -157 a - 64\) , \( 845 a + 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-157a-64\right){x}+845a+50$ |
| 21609.3-a7 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{20} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B[2] |
$1$ |
\( 2^{6} \) |
$1$ |
$0.279582893$ |
1.291338071 |
\( -\frac{854150427}{117649} a + \frac{1556711379}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 63 a + 891\) , \( 11943 a - 6308\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(63a+891\right){x}+11943a-6308$ |
| 21609.3-a8 |
21609.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{19} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{5} \) |
$1$ |
$0.139791446$ |
1.291338071 |
\( -\frac{308817493407}{2401} a + \frac{246921503922}{2401} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 798 a + 14121\) , \( 780753 a - 436871\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(798a+14121\right){x}+780753a-436871$ |
| 21609.3-b1 |
21609.3-b |
$2$ |
$13$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{32} \cdot 7^{10} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 13$ |
2Cn, 13B.4.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.185208367$ |
$0.090987281$ |
2.988518917 |
\( -\frac{1713910976512}{1594323} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -8210 a - 13685\) , \( -646946 a - 543199\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8210a-13685\right){x}-646946a-543199$ |
| 21609.3-b2 |
21609.3-b |
$2$ |
$13$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{8} \cdot 7^{10} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 13$ |
2Cn, 13B.4.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.091169874$ |
$1.182834662$ |
2.988518917 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -20 a - 35\) , \( 64 a + 71\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-20a-35\right){x}+64a+71$ |
| 21609.3-c1 |
21609.3-c |
$2$ |
$13$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{32} \cdot 7^{10} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 13$ |
2Cn, 13B.4.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.185208367$ |
$0.090987281$ |
2.988518917 |
\( -\frac{1713910976512}{1594323} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 13686 a + 8211\) , \( 625050 a - 1176460\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(13686a+8211\right){x}+625050a-1176460$ |
| 21609.3-c2 |
21609.3-c |
$2$ |
$13$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{8} \cdot 7^{10} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 13$ |
2Cn, 13B.4.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.091169874$ |
$1.182834662$ |
2.988518917 |
\( -\frac{28672}{3} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 36 a + 21\) , \( -120 a + 170\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(36a+21\right){x}-120a+170$ |
| 21609.3-d1 |
21609.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{12} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$1.078979362$ |
2.491796100 |
\( -3375 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -53 a + 33\) , \( -79 a + 162\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-53a+33\right){x}-79a+162$ |
| 21609.3-d2 |
21609.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{12} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$1.078979362$ |
2.491796100 |
\( -3375 \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -34 a + 53\) , \( 78 a + 83\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-34a+53\right){x}+78a+83$ |
| 21609.3-d3 |
21609.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{12} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{4} \) |
$1$ |
$0.539489681$ |
2.491796100 |
\( 16581375 \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -559 a + 893\) , \( 5013 a + 4535\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-559a+893\right){x}+5013a+4535$ |
| 21609.3-d4 |
21609.3-d |
$4$ |
$14$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
21609.3 |
\( 3^{2} \cdot 7^{4} \) |
\( 3^{6} \cdot 7^{12} \) |
$1.87654$ |
$(-2a+1), (-3a+1), (3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{4} \) |
$1$ |
$0.539489681$ |
2.491796100 |
\( 16581375 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -893 a + 558\) , \( -5014 a + 9549\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-893a+558\right){x}-5014a+9549$ |
*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.
