| 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 |
| 1089.2-a1 |
1089.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{16} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.160860264$ |
3.649847049 |
\( -\frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 3816 a - 6611\) , \( 72469 a - 125520\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3816a-6611\right){x}+72469a-125520$ |
| 1089.2-a2 |
1089.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{8} \cdot 11^{11} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$25$ |
\( 2^{4} \) |
$1$ |
$0.126434410$ |
3.649847049 |
\( -\frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 383820452 a - 664796525\) , \( 5386969633888 a - 9330505104725\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(383820452a-664796525\right){x}+5386969633888a-9330505104725$ |
| 1089.2-a3 |
1089.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{7} \cdot 11^{16} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$25$ |
\( 2^{4} \) |
$1$ |
$0.126434410$ |
3.649847049 |
\( \frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 6889231 a - 11932502\) , \( 12956098743 a - 22440621292\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6889231a-11932502\right){x}+12956098743a-22440621292$ |
| 1089.2-a4 |
1089.2-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{11} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$3.160860264$ |
3.649847049 |
\( \frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -253 a + 412\) , \( 1249 a - 2108\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-253a+412\right){x}+1249a-2108$ |
| 1089.2-b1 |
1089.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$6.230339980$ |
3.597088464 |
\( \frac{225280}{9} a - \frac{630784}{9} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -55 a - 153\) , \( 452 a + 923\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(-55a-153\right){x}+452a+923$ |
| 1089.2-c1 |
1089.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.893018983$ |
1.092935019 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 399 a - 692\bigr] \) |
${y}^2+{y}={x}^{3}+399a-692$ |
| 1089.2-c2 |
1089.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.679056949$ |
1.092935019 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -399 a + 691\bigr] \) |
${y}^2+a{y}={x}^{3}-399a+691$ |
| 1089.2-d1 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$2.399462645$ |
$1.714750314$ |
2.375495747 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 2401 a - 4161\) , \( 85004 a - 147232\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2401a-4161\right){x}+85004a-147232$ |
| 1089.2-d2 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.266606960$ |
$15.43275283$ |
2.375495747 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 2401 a - 4160\) , \( -85004 a + 147231\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(2401a-4160\right){x}-85004a+147231$ |
| 1089.2-d3 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{9} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.799820881$ |
$5.144250944$ |
2.375495747 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 388 a - 672\) , \( -336 a + 582\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(388a-672\right){x}-336a+582$ |
| 1089.2-d4 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{9} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{3} \) |
$0.399910440$ |
$5.144250944$ |
2.375495747 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -29 a + 49\) , \( 24 a - 43\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-29a+49\right){x}+24a-43$ |
| 1089.2-d5 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1.199731322$ |
$1.714750314$ |
2.375495747 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 38 a - 80\) , \( 219 a - 394\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(38a-80\right){x}+219a-394$ |
| 1089.2-d6 |
1089.2-d |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{3} \cdot 11^{6} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.133303480$ |
$15.43275283$ |
2.375495747 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 38 a - 81\) , \( -219 a + 393\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(38a-81\right){x}-219a+393$ |
| 1089.2-e1 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( -\frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -24 a - 39\) , \( 105 a + 181\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-24a-39\right){x}+105a+181$ |
| 1089.2-e2 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{14} \cdot 11^{14} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.289582703$ |
1.337525213 |
\( \frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 1041 a + 1491\) , \( 29613 a + 48754\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(1041a+1491\right){x}+29613a+48754$ |
| 1089.2-e3 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{10} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$1.158330812$ |
1.337525213 |
\( -\frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -324 a - 879\) , \( 5553 a + 7159\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-324a-879\right){x}+5553a+7159$ |
| 1089.2-e4 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.289582703$ |
1.337525213 |
\( -\frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -729 a - 6369\) , \( -27531 a - 192416\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-729a-6369\right){x}-27531a-192416$ |
| 1089.2-e5 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( \frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -384 a - 684\) , \( 5634 a + 9709\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-384a-684\right){x}+5634a+9709$ |
| 1089.2-e6 |
1089.2-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.633323250$ |
1.337525213 |
\( \frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -6204 a - 10809\) , \( 353571 a + 612511\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-6204a-10809\right){x}+353571a+612511$ |
| 1089.2-f1 |
1089.2-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{16} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( -\frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 3816 a - 6612\) , \( -72469 a + 125519\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(3816a-6612\right){x}-72469a+125519$ |
| 1089.2-f2 |
1089.2-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{8} \cdot 11^{11} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( -\frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 383820452 a - 664796526\) , \( -5386969633888 a + 9330505104724\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(383820452a-664796526\right){x}-5386969633888a+9330505104724$ |
| 1089.2-f3 |
1089.2-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{7} \cdot 11^{16} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( \frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 6889231 a - 11932503\) , \( -12956098743 a + 22440621291\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(6889231a-11932503\right){x}-12956098743a+22440621291$ |
| 1089.2-f4 |
1089.2-f |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{11} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$1.529686561$ |
0.883164947 |
\( \frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -253 a + 411\) , \( -1249 a + 2107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-253a+411\right){x}-1249a+2107$ |
| 1089.2-g1 |
1089.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{6} \cdot 11^{3} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.329680234$ |
$12.32964666$ |
2.346836933 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -8 a - 12\) , \( -6 a - 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a-12\right){x}-6a-12$ |
| 1089.2-g2 |
1089.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( - 3^{6} \cdot 11^{3} \) |
$1.77822$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.164840117$ |
$12.32964666$ |
2.346836933 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( a + 1\) , \( 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+1\right){x}+2$ |
| 1089.2-h1 |
1089.2-h |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.516613227$ |
0.894800358 |
\( \frac{225280}{9} a - \frac{630784}{9} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -55 a - 153\) , \( -452 a - 924\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-55a-153\right){x}-452a-924$ |
| 1089.2-i1 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( -\frac{28016}{33} a - \frac{15365}{11} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -24 a - 38\) , \( -106 a - 183\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-24a-38\right){x}-106a-183$ |
| 1089.2-i2 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{14} \cdot 11^{14} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.672647213$ |
0.776706099 |
\( \frac{66041766161825}{17363069361} a - \frac{104139369666842}{17363069361} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 1041 a + 1492\) , \( -29614 a - 48756\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1041a+1492\right){x}-29614a-48756$ |
| 1089.2-i3 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{10} \cdot 11^{10} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( -\frac{6743741507300}{131769} a + \frac{11681077261807}{131769} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -324 a - 878\) , \( -5554 a - 7161\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-324a-878\right){x}-5554a-7161$ |
| 1089.2-i4 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( -\frac{3293747382143872955}{363} a + \frac{5704937813176114478}{363} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -729 a - 6368\) , \( 27530 a + 192414\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-729a-6368\right){x}+27530a+192414$ |
| 1089.2-i5 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{8} \cdot 11^{8} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.690588852$ |
0.776706099 |
\( \frac{1324878680}{363} a + \frac{2299866043}{363} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -384 a - 683\) , \( -5635 a - 9711\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-384a-683\right){x}-5635a-9711$ |
| 1089.2-i6 |
1089.2-i |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1089.2 |
\( 3^{2} \cdot 11^{2} \) |
\( 3^{7} \cdot 11^{7} \) |
$1.77822$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.672647213$ |
0.776706099 |
\( \frac{1526015049596036}{33} a + \frac{881045199725315}{11} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -6204 a - 10808\) , \( -353572 a - 612513\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6204a-10808\right){x}-353572a-612513$ |
*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.
