| 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 |
| 484.2-a1 |
484.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{10} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$5.235137245$ |
1.511253948 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 1\) , \( 31 a + 66\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+{x}+31a+66$ |
| 484.2-a2 |
484.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{10} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1$ |
$1.745045748$ |
1.511253948 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 1\) , \( -32 a - 68\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+{x}-32a-68$ |
| 484.2-b1 |
484.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{8} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$7.784606037$ |
2.247222195 |
\( -301117383680 a - 521550665728 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( 637 a - 1214\) , \( -13086 a + 23086\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(637a-1214\right){x}-13086a+23086$ |
| 484.2-b2 |
484.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{8} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$7.784606037$ |
2.247222195 |
\( -3072 a - 4096 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -33 a + 56\) , \( -54 a + 92\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-33a+56\right){x}-54a+92$ |
| 484.2-c1 |
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 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.240929030$ |
1.333813215 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( 43 a + 75\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}+43a+75$ |
| 484.2-c2 |
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 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.080309676$ |
1.333813215 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( -43 a - 75\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+{x}-43a-75$ |
| 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 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 110 a - 206\) , \( 916 a - 1570\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(110a-206\right){x}+916a-1570$ |
| 484.2-c5 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$18.48185806$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 5 a - 16\) , \( -14 a + 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(5a-16\right){x}-14a+30$ |
| 484.2-c6 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.160619353$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 5 a - 16\) , \( 14 a - 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a-16\right){x}+14a-30$ |
| 484.2-c7 |
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 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -20 a - 86\) , \( 154 a + 210\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-20a-86\right){x}+154a+210$ |
| 484.2-c8 |
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 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -20 a - 86\) , \( -154 a - 210\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-20a-86\right){x}-154a-210$ |
| 484.2-d1 |
484.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{8} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$0.252776699$ |
0.656733130 |
\( -301117383680 a - 521550665728 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 637 a - 1214\) , \( 13085 a - 23088\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(637a-1214\right){x}+13085a-23088$ |
| 484.2-d2 |
484.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{8} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.274990299$ |
0.656733130 |
\( -3072 a - 4096 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -33 a + 56\) , \( 53 a - 94\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-33a+56\right){x}+53a-94$ |
*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.
