| 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 |
| 288.1-a1 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{12} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( -\frac{166016}{3} a + 95936 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 4063804 a - 7038714\) , \( -5839863922 a + 10114941022\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(4063804a-7038714\right){x}-5839863922a+10114941022$ |
| 288.1-a2 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.510861184$ |
1.449646380 |
\( -\frac{164847992914}{3} a + \frac{285525100658}{3} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 381758 a - 661224\) , \( 168866036 a - 292484554\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(381758a-661224\right){x}+168866036a-292484554$ |
| 288.1-a3 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{10} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( -\frac{1122088}{9} a + \frac{1989808}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 23873 a - 41349\) , \( 2630312 a - 4555834\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(23873a-41349\right){x}+2630312a-4555834$ |
| 288.1-a4 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( \frac{9856}{3} a + \frac{22336}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 113682 a - 196902\) , \( 3469554 a - 6009444\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(113682a-196902\right){x}+3469554a-6009444$ |
| 288.1-a5 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{14} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.021722368$ |
1.449646380 |
\( \frac{443186854}{81} a + \frac{767608522}{81} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 77 a - 140\) , \( 1663 a - 2881\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(77a-140\right){x}+1663a-2881$ |
| 288.1-a6 |
288.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{6} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.021722368$ |
1.449646380 |
\( \frac{132636728}{3} a + 76579552 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -1569806 a + 2718984\) , \( 176681708 a - 306021695\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-1569806a+2718984\right){x}+176681708a-306021695$ |
| 288.1-b1 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{14} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.021722368$ |
1.449646380 |
\( -\frac{443186854}{81} a + \frac{767608522}{81} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -79 a - 140\) , \( -1664 a - 2881\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-79a-140\right){x}-1664a-2881$ |
| 288.1-b2 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{6} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.021722368$ |
1.449646380 |
\( -\frac{132636728}{3} a + 76579552 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 10822 a - 18744\) , \( 809620 a - 1402303\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10822a-18744\right){x}+809620a-1402303$ |
| 288.1-b3 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( -\frac{9856}{3} a + \frac{22336}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 198 a - 342\) , \( 1746 a - 3024\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(198a-342\right){x}+1746a-3024$ |
| 288.1-b4 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{12} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( \frac{166016}{3} a + 95936 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -668 a + 1158\) , \( 54226 a - 93922\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-668a+1158\right){x}+54226a-93922$ |
| 288.1-b5 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{10} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.04344473$ |
1.449646380 |
\( \frac{1122088}{9} a + \frac{1989808}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 11 a - 21\) , \( -20 a + 34\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11a-21\right){x}-20a+34$ |
| 288.1-b6 |
288.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.510861184$ |
1.449646380 |
\( \frac{164847992914}{3} a + \frac{285525100658}{3} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -34 a + 24\) , \( -164 a + 214\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-34a+24\right){x}-164a+214$ |
| 288.1-c1 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{12} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.410195878$ |
$7.783091503$ |
1.843243885 |
\( -\frac{166016}{3} a + 95936 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 4063804 a - 7038714\) , \( 5839863922 a - 10114941022\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(4063804a-7038714\right){x}+5839863922a-10114941022$ |
| 288.1-c2 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.820391757$ |
$7.783091503$ |
1.843243885 |
\( -\frac{164847992914}{3} a + \frac{285525100658}{3} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 381758 a - 661224\) , \( -168866036 a + 292484554\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(381758a-661224\right){x}-168866036a+292484554$ |
| 288.1-c3 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{10} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.410195878$ |
$15.56618300$ |
1.843243885 |
\( -\frac{1122088}{9} a + \frac{1989808}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 23873 a - 41349\) , \( -2630312 a + 4555834\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(23873a-41349\right){x}-2630312a+4555834$ |
| 288.1-c4 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.205097939$ |
$15.56618300$ |
1.843243885 |
\( \frac{9856}{3} a + \frac{22336}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 113682 a - 196902\) , \( -3469554 a + 6009444\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(113682a-196902\right){x}-3469554a+6009444$ |
| 288.1-c5 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{14} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.820391757$ |
$3.891545751$ |
1.843243885 |
\( \frac{443186854}{81} a + \frac{767608522}{81} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 77 a - 140\) , \( -1664 a + 2879\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(77a-140\right){x}-1664a+2879$ |
| 288.1-c6 |
288.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{6} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.410195878$ |
$15.56618300$ |
1.843243885 |
\( \frac{132636728}{3} a + 76579552 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -1569806 a + 2718984\) , \( -176681708 a + 306021695\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1569806a+2718984\right){x}-176681708a+306021695$ |
| 288.1-d1 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{14} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.820391757$ |
$3.891545751$ |
1.843243885 |
\( -\frac{443186854}{81} a + \frac{767608522}{81} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -79 a - 140\) , \( 1663 a + 2879\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-79a-140\right){x}+1663a+2879$ |
| 288.1-d2 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{6} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.410195878$ |
$15.56618300$ |
1.843243885 |
\( -\frac{132636728}{3} a + 76579552 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 10822 a - 18744\) , \( -809620 a + 1402303\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(10822a-18744\right){x}-809620a+1402303$ |
| 288.1-d3 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.205097939$ |
$15.56618300$ |
1.843243885 |
\( -\frac{9856}{3} a + \frac{22336}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 198 a - 342\) , \( -1746 a + 3024\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(198a-342\right){x}-1746a+3024$ |
| 288.1-d4 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{12} \cdot 3^{7} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.410195878$ |
$7.783091503$ |
1.843243885 |
\( \frac{166016}{3} a + 95936 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -668 a + 1158\) , \( -54226 a + 93922\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-668a+1158\right){x}-54226a+93922$ |
| 288.1-d5 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{10} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.410195878$ |
$15.56618300$ |
1.843243885 |
\( \frac{1122088}{9} a + \frac{1989808}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 11 a - 21\) , \( 20 a - 34\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(11a-21\right){x}+20a-34$ |
| 288.1-d6 |
288.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
288.1 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{9} \cdot 3^{8} \) |
$1.27520$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.820391757$ |
$7.783091503$ |
1.843243885 |
\( \frac{164847992914}{3} a + \frac{285525100658}{3} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -34 a + 24\) , \( 164 a - 214\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-34a+24\right){x}+164a-214$ |
*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.
