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-a1 |
16.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$2.55261$ |
$(a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.054516910$ |
$18.51924189$ |
5.327799054 |
\( -27648 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 16\) , \( a - 13\bigr] \) |
${y}^2+\left(w+1\right){y}={x}^3+w{x}^2+16{x}+w-13$ |
16.1-b1 |
16.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$2.55261$ |
$(a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.054516910$ |
$18.51924189$ |
5.327799054 |
\( -27648 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 16\) , \( -2 a - 13\bigr] \) |
${y}^2+\left(w+1\right){y}={x}^3-w{x}^2+16{x}-2w-13$ |
16.1-c1 |
16.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.55261$ |
$(a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.595981725$ |
$18.51924189$ |
1.545507295 |
\( -27648 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -9\) , \( -2 a - 13\bigr] \) |
${y}^2+\left(w+1\right){y}={x}^3-9{x}-2w-13$ |
16.1-d1 |
16.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.55261$ |
$(a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.595981725$ |
$18.51924189$ |
1.545507295 |
\( -27648 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -9\) , \( a - 13\bigr] \) |
${y}^2+\left(w+1\right){y}={x}^3-9{x}+w-13$ |
17.1-a1 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{8} \) |
$2.59159$ |
$(17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$7.539083304$ |
4.222731281 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( 377\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-6{x}+377$ |
17.1-a2 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$2.59159$ |
$(17,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$30.15633321$ |
4.222731281 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-6{x}-1$ |
17.1-a3 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{4} \) |
$2.59159$ |
$(17,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2 \) |
$1$ |
$30.15633321$ |
4.222731281 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -51\) , \( 152\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-51{x}+152$ |
17.1-a4 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$2.59159$ |
$(17,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$30.15633321$ |
4.222731281 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -816\) , \( 9179\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-816{x}+9179$ |
17.1-b1 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{8} \) |
$2.59159$ |
$(17,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$11.00049651$ |
$2.393455763$ |
3.686825688 |
\( -\frac{35937}{83521} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 48\) , \( -325\bigr] \) |
${y}^2+w{x}{y}={x}^3+48{x}-325$ |
17.1-b2 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.750124128$ |
$38.29529222$ |
3.686825688 |
\( \frac{35937}{17} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 48\) , \( 53\bigr] \) |
${y}^2+w{x}{y}={x}^3+48{x}+53$ |
17.1-b3 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{4} \) |
$2.59159$ |
$(17,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$11.00049651$ |
$9.573823055$ |
3.686825688 |
\( \frac{20346417}{289} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 3\) , \( -280\bigr] \) |
${y}^2+w{x}{y}={x}^3+3{x}-280$ |
17.1-b4 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$11.00049651$ |
$2.393455763$ |
3.686825688 |
\( \frac{82483294977}{17} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -762\) , \( -12367\bigr] \) |
${y}^2+w{x}{y}={x}^3-762{x}-12367$ |
17.1-c1 |
17.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.036297434$ |
$7.539083304$ |
1.074842109 |
\( -\frac{35937}{83521} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 28\) , \( 75\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3+28{x}+75$ |
17.1-c2 |
17.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.509074358$ |
$30.15633321$ |
1.074842109 |
\( \frac{35937}{17} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 28\) , \( 61\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3+28{x}+61$ |
17.1-c3 |
17.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.018148717$ |
$30.15633321$ |
1.074842109 |
\( \frac{20346417}{289} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 23\) , \( 45\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3+23{x}+45$ |
17.1-c4 |
17.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.036297434$ |
$30.15633321$ |
1.074842109 |
\( \frac{82483294977}{17} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -62\) , \( 11\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3-62{x}+11$ |
17.1-d1 |
17.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$13.65757227$ |
$2.393455763$ |
2.288673435 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( -14\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-{x}-14$ |
17.1-d2 |
17.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.414393067$ |
$38.29529222$ |
2.288673435 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-{x}$ |
17.1-d3 |
17.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$6.828786135$ |
$9.573823055$ |
2.288673435 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-6{x}-4$ |
17.1-d4 |
17.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$2.59159$ |
$(17,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$13.65757227$ |
$2.393455763$ |
2.288673435 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -91\) , \( -310\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-91{x}-310$ |
24.1-a1 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{28} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$2.540027220$ |
$2.325279868$ |
6.616350463 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 24687 a + 176301\) , \( -41091534 a - 293452249\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(w-1\right){x}^2+\left(24687w+176301\right){x}-41091534w-293452249$ |
24.1-a2 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.270013610$ |
$18.60223895$ |
6.616350463 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \) |
${y}^2={x}^3+6{x}+7$ |
24.1-a3 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.540027220$ |
$37.20447790$ |
6.616350463 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -6813 a - 48654\) , \( 560799 a + 4004906\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(w-1\right){x}^2+\left(-6813w-48654\right){x}+560799w+4004906$ |
24.1-a4 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{20} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.080054440$ |
$9.301119475$ |
6.616350463 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 38336 a - 273635\) , \( 10427941 a - 74469977\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(38336w-273635\right){x}+10427941w-74469977$ |
24.1-a5 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.080054440$ |
$37.20447790$ |
6.616350463 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -101313 a - 723519\) , \( 46145736 a + 329546471\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(w-1\right){x}^2+\left(-101313w-723519\right){x}+46145736w+329546471$ |
24.1-a6 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$10.16010888$ |
$2.325279868$ |
6.616350463 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 605336 a - 4322825\) , \( 685333873 a - 4894262387\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(605336w-4322825\right){x}+685333873w-4894262387$ |
24.1-b1 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$5.683508517$ |
0.795850378 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2746 a + 19640\) , \( -1543982 a + 11026268\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3-w{x}^2+\left(-2746w+19640\right){x}-1543982w+11026268$ |
24.1-b2 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.36701703$ |
0.795850378 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3+{x}^2+{x}$ |
24.1-b3 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
0.795850378 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -754 a - 5355\) , \( -26862 a - 191802\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-754w-5355\right){x}-26862w-191802$ |
24.1-b4 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$22.73403407$ |
0.795850378 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 4254 a - 30350\) , \( -362098 a + 2585928\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3-w{x}^2+\left(4254w-30350\right){x}-362098w+2585928$ |
24.1-b5 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$5.683508517$ |
0.795850378 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -11254 a - 80340\) , \( -1799688 a - 12852312\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-11254w-80340\right){x}-1799688w-12852312$ |
24.1-b6 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
0.795850378 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -67254 a - 480260\) , \( 25001614 a + 178547268\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-67254w-480260\right){x}+25001614w+178547268$ |
24.1-c1 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{28} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1.492303742$ |
$5.683508517$ |
4.750601994 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 24686 a + 176275\) , \( 41465209 a + 296120813\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(24686w+176275\right){x}+41465209w+296120813$ |
24.1-c2 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$11.93842994$ |
$11.36701703$ |
4.750601994 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^3+6{x}-7$ |
24.1-c3 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$5.969214971$ |
$22.73403407$ |
4.750601994 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -6814 a - 48680\) , \( -664079 a - 4742482\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(-6814w-48680\right){x}-664079w-4742482$ |
24.1-c4 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{20} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.984607485$ |
$22.73403407$ |
4.750601994 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 38337 a - 273609\) , \( -10394976 a + 74235411\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(w-1\right){x}^2+\left(38337w-273609\right){x}-10394976w+74235411$ |
24.1-c5 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.984607485$ |
$5.683508517$ |
4.750601994 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -101314 a - 723545\) , \( -47679881 a - 340502467\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(-101314w-723545\right){x}-47679881w-340502467$ |
24.1-c6 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.492303742$ |
$22.73403407$ |
4.750601994 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 605337 a - 4322799\) , \( -684814098 a + 4890551301\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(w-1\right){x}^2+\left(605337w-4322799\right){x}-684814098w+4890551301$ |
24.1-d1 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.325279868$ |
0.651208618 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -2738 a + 19623\) , \( 1519308 a - 10849845\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+{x}^2+\left(-2738w+19623\right){x}+1519308w-10849845$ |
24.1-d2 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$18.60223895$ |
0.651208618 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}^2+{x}$ |
24.1-d3 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.651208618 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -764 a - 5372\) , \( 20035 a + 143270\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(-w+1\right){x}^2+\left(-764w-5372\right){x}+20035w+143270$ |
24.1-d4 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$9.301119475$ |
0.651208618 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 4262 a - 30367\) , \( 400424 a - 2859415\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+{x}^2+\left(4262w-30367\right){x}+400424w-2859415$ |
24.1-d5 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.651208618 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -11264 a - 80357\) , \( 1698361 a + 12128915\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(-w+1\right){x}^2+\left(-11264w-80357\right){x}+1698361w+12128915$ |
24.1-d6 |
24.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$2.82492$ |
$(a-7), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.325279868$ |
0.651208618 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -67264 a - 480277\) , \( -25606941 a - 182869945\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3+\left(-w+1\right){x}^2+\left(-67264w-480277\right){x}-25606941w-182869945$ |
25.2-a1 |
25.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 3^{12} \cdot 5^{10} \) |
$2.85390$ |
$(5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
0.514337188 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 24 a + 155\bigr] \) |
${y}^2+{y}={x}^3+24w+155$ |
25.2-a2 |
25.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{10} \) |
$2.85390$ |
$(5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
0.514337188 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 17\) , \( a - 6\bigr] \) |
${y}^2+{y}={x}^3+w{x}^2+17{x}+w-6$ |
25.2-b1 |
25.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{10} \) |
$2.85390$ |
$(5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$7.346204439$ |
4.629034696 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 17\) , \( -a - 7\bigr] \) |
${y}^2+w{y}={x}^3-w{x}^2+17{x}-w-7$ |
25.2-b2 |
25.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 3^{12} \cdot 5^{10} \) |
$2.85390$ |
$(5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$7.346204439$ |
4.629034696 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -24 a - 168\bigr] \) |
${y}^2+w{y}={x}^3-24w-168$ |
25.3-a1 |
25.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$2.85390$ |
$(5,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
0.514337188 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 17\) , \( -a - 6\bigr] \) |
${y}^2+{y}={x}^3-w{x}^2+17{x}-w-6$ |
25.3-a2 |
25.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{51}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 3^{12} \cdot 5^{10} \) |
$2.85390$ |
$(5,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
0.514337188 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -24 a + 155\bigr] \) |
${y}^2+{y}={x}^3-24w+155$ |
*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.