| 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 |
| 16.1-a3 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$8.847515954$ |
0.638514464 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 13\) , \( 11 a - 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-13\right){x}+11a-21$ |
| 16.1-a4 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 4 a - 13\) , \( -12 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-13\right){x}-12a+19$ |
| 36.1-a3 |
36.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$0.75824$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$5.898343969$ |
0.851352619 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 239 a - 416\) , \( 2458 a - 4259\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(239a-416\right){x}+2458a-4259$ |
| 36.1-a4 |
36.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$0.75824$ |
$(a+1), (a)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$17.69503190$ |
0.851352619 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 240 a - 414\) , \( -2874 a + 4978\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(240a-414\right){x}-2874a+4978$ |
| 256.1-c3 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$4.423757977$ |
1.277028929 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 20 a - 44\) , \( 92 a - 160\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(20a-44\right){x}+92a-160$ |
| 256.1-c4 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 20 a - 44\) , \( -92 a + 160\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(20a-44\right){x}-92a+160$ |
| 484.2-c3 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$9.240929030$ |
1.333813215 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 110 a - 206\) , \( -916 a + 1570\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(110a-206\right){x}-916a+1570$ |
| 484.2-c4 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.080309676$ |
1.333813215 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 110 a - 206\) , \( 916 a - 1570\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(110a-206\right){x}+916a-1570$ |
| 484.3-c3 |
484.3-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$9.240929030$ |
1.333813215 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 6769 a - 11728\) , \( -397102 a + 687799\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6769a-11728\right){x}-397102a+687799$ |
| 484.3-c4 |
484.3-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.080309676$ |
1.333813215 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 6769 a - 11728\) , \( 397101 a - 687801\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(6769a-11728\right){x}+397101a-687801$ |
| 676.2-a3 |
676.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.845874701$ |
$9.815437671$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 2630 a - 4556\) , \( -95878 a + 166066\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2630a-4556\right){x}-95878a+166066$ |
| 676.2-a4 |
676.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$1.127832935$ |
$2.453859417$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 2630 a - 4556\) , \( 95878 a - 166066\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(2630a-4556\right){x}+95878a-166066$ |
| 676.3-a3 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.383498806$ |
$2.453859417$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 409 a - 718\) , \( 5865 a - 10165\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(409a-718\right){x}+5865a-10165$ |
| 676.3-a4 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.281958233$ |
$9.815437671$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 409 a - 718\) , \( -5866 a + 10163\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(409a-718\right){x}-5866a+10163$ |
| 1024.1-j3 |
1024.1-j |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{26} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.429957487$ |
$10.83594978$ |
2.689873605 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -10 a - 59\) , \( -49 a - 2\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-10a-59\right){x}-49a-2$ |
| 1024.1-j4 |
1024.1-j |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{26} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.289872463$ |
$3.611983263$ |
2.689873605 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -10 a - 59\) , \( 49 a + 2\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-10a-59\right){x}+49a+2$ |
| 1024.1-k3 |
1024.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{26} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.289872463$ |
$3.611983263$ |
2.689873605 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 170 a - 299\) , \( 1711 a - 2958\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(170a-299\right){x}+1711a-2958$ |
| 1024.1-k4 |
1024.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{26} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.429957487$ |
$10.83594978$ |
2.689873605 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 170 a - 299\) , \( -1711 a + 2958\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(170a-299\right){x}-1711a+2958$ |
| 2304.1-v3 |
2304.1-v |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{6} \) |
$2.14462$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.653234677$ |
$8.847515954$ |
3.336798323 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -120 a - 225\) , \( -510 a - 856\bigr] \) |
${y}^2={x}^{3}+\left(-120a-225\right){x}-510a-856$ |
| 2304.1-v4 |
2304.1-v |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{6} \) |
$2.14462$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.959704032$ |
$2.949171984$ |
3.336798323 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -120 a - 225\) , \( 510 a + 856\bigr] \) |
${y}^2={x}^{3}+\left(-120a-225\right){x}+510a+856$ |
*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.
