| 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 |
| 1089.3-a1 |
1089.3-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{7} \cdot 11^{16} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$25$ |
\( 2^{4} \) |
$1$ |
$0.126434410$ |
3.649847049 |
\( -\frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 29876 a - 52382\) , \( 3718637 a - 6447198\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(29876a-52382\right){x}+3718637a-6447198$ |
| 1089.3-a2 |
1089.3-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{11} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.160860264$ |
3.649847049 |
\( -\frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 252 a + 412\) , \( -1249 a - 2108\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(252a+412\right){x}-1249a-2108$ |
| 1089.3-a3 |
1089.3-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{16} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.160860264$ |
3.649847049 |
\( \frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( a - 71\) , \( -159 a + 164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(a-71\right){x}-159a+164$ |
| 1089.3-a4 |
1089.3-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{8} \cdot 11^{11} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$25$ |
\( 2^{4} \) |
$1$ |
$0.126434410$ |
3.649847049 |
\( \frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 75657 a - 131225\) , \( 37907402 a - 65658545\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(75657a-131225\right){x}+37907402a-65658545$ |
| 1089.3-b1 |
1089.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$6.230339980$ |
3.597088464 |
\( -\frac{225280}{9} a - \frac{630784}{9} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 55 a - 153\) , \( -452 a + 923\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(55a-153\right){x}-452a+923$ |
| 1089.3-c1 |
1089.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.893018983$ |
1.092935019 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -399 a - 692\bigr] \) |
${y}^2+{y}={x}^{3}-399a-692$ |
| 1089.3-c2 |
1089.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.679056949$ |
1.092935019 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 399 a + 691\bigr] \) |
${y}^2+a{y}={x}^{3}+399a+691$ |
| 1089.3-d1 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.133303480$ |
$15.43275283$ |
2.375495747 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 731 a - 1269\) , \( -13968 a + 24193\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(731a-1269\right){x}-13968a+24193$ |
| 1089.3-d2 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1.199731322$ |
$1.714750314$ |
2.375495747 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 731 a - 1268\) , \( 13968 a - 24194\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(731a-1268\right){x}+13968a-24194$ |
| 1089.3-d3 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{9} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.799820881$ |
$5.144250944$ |
2.375495747 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -11 a + 13\) , \( 6 a - 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-11a+13\right){x}+6a-16$ |
| 1089.3-d4 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{9} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{3} \) |
$0.399910440$ |
$5.144250944$ |
2.375495747 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 118 a - 204\) , \( -102 a + 177\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(118a-204\right){x}-102a+177$ |
| 1089.3-d5 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.266606960$ |
$15.43275283$ |
2.375495747 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -44\) , \( 3 a + 115\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}-44{x}+3a+115$ |
| 1089.3-d6 |
1089.3-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$2.399462645$ |
$1.714750314$ |
2.375495747 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -45\) , \( -3 a - 116\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}-45{x}-3a-116$ |
| 1089.3-e1 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{14} \cdot 11^{14} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.289582703$ |
1.337525213 |
\( -\frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -1042 a + 1491\) , \( -29614 a + 48754\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-1042a+1491\right){x}-29614a+48754$ |
| 1089.3-e2 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( \frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 4397 a - 7616\) , \( -268106 a + 464373\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4397a-7616\right){x}-268106a+464373$ |
| 1089.3-e3 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( -\frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -3596 a - 6338\) , \( 124411 a + 216028\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3596a-6338\right){x}+124411a+216028$ |
| 1089.3-e4 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( -\frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 383 a - 684\) , \( -5635 a + 9709\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(383a-684\right){x}-5635a+9709$ |
| 1089.3-e5 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{10} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$1.158330812$ |
1.337525213 |
\( \frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 323 a - 879\) , \( -5554 a + 7159\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(323a-879\right){x}-5554a+7159$ |
| 1089.3-e6 |
1089.3-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.289582703$ |
1.337525213 |
\( \frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 728 a - 6369\) , \( 27530 a - 192416\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(728a-6369\right){x}+27530a-192416$ |
| 1089.3-f1 |
1089.3-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{7} \cdot 11^{16} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( -\frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 29876 a - 52383\) , \( -3718637 a + 6447197\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(29876a-52383\right){x}-3718637a+6447197$ |
| 1089.3-f2 |
1089.3-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{11} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( -\frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 252 a + 411\) , \( 1249 a + 2107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(252a+411\right){x}+1249a+2107$ |
| 1089.3-f3 |
1089.3-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{16} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( \frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( a - 72\) , \( 159 a - 165\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a-72\right){x}+159a-165$ |
| 1089.3-f4 |
1089.3-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{8} \cdot 11^{11} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( \frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 75657 a - 131226\) , \( -37907402 a + 65658544\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(75657a-131226\right){x}-37907402a+65658544$ |
| 1089.3-g1 |
1089.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{6} \cdot 11^{3} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.329680234$ |
$12.32964666$ |
2.346836933 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 7 a - 11\) , \( -6 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(7a-11\right){x}-6a+11$ |
| 1089.3-g2 |
1089.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{6} \cdot 11^{3} \) |
$1.77822$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.164840117$ |
$12.32964666$ |
2.346836933 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -104 a + 180\) , \( 90 a - 156\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-104a+180\right){x}+90a-156$ |
| 1089.3-h1 |
1089.3-h |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.516613227$ |
0.894800358 |
\( -\frac{225280}{9} a - \frac{630784}{9} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 55 a - 153\) , \( 452 a - 924\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(55a-153\right){x}+452a-924$ |
| 1089.3-i1 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{14} \cdot 11^{14} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.672647213$ |
0.776706099 |
\( -\frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -1042 a + 1492\) , \( 29613 a - 48756\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1042a+1492\right){x}+29613a-48756$ |
| 1089.3-i2 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( \frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 4397 a - 7617\) , \( 268106 a - 464374\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(4397a-7617\right){x}+268106a-464374$ |
| 1089.3-i3 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.672647213$ |
0.776706099 |
\( -\frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3596 a - 6339\) , \( -124411 a - 216029\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3596a-6339\right){x}-124411a-216029$ |
| 1089.3-i4 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( -\frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 383 a - 683\) , \( 5634 a - 9711\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(383a-683\right){x}+5634a-9711$ |
| 1089.3-i5 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{10} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( \frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 323 a - 878\) , \( 5553 a - 7161\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(323a-878\right){x}+5553a-7161$ |
| 1089.3-i6 |
1089.3-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.3 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( \frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 728 a - 6368\) , \( -27531 a + 192414\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(728a-6368\right){x}-27531a+192414$ |
*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.
