| 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 |
| 256.1-a1 |
256.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.75994948$ |
2.784449256 |
\( -10220 a - 17724 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -3 a - 4\) , \( 5 a + 8\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-3a-4\right){x}+5a+8$ |
| 256.1-a2 |
256.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.75994948$ |
2.784449256 |
\( 481853450 a + 863142628 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -43 a - 84\) , \( 261 a + 464\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-43a-84\right){x}+261a+464$ |
| 256.1-b1 |
256.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$2.329446770$ |
$2.472252300$ |
2.513424990 |
\( -3375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 15\) , \( 12 a - 34\bigr] \) |
${y}^2={x}^{3}+\left(5a-15\right){x}+12a-34$ |
| 256.1-b2 |
256.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.332778110$ |
$17.30576610$ |
2.513424990 |
\( -3375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -5 a - 10\) , \( 12 a + 22\bigr] \) |
${y}^2={x}^{3}+\left(-5a-10\right){x}+12a+22$ |
| 256.1-b3 |
256.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$4.658893541$ |
$2.472252300$ |
2.513424990 |
\( 16581375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 85 a - 255\) , \( 684 a - 1938\bigr] \) |
${y}^2={x}^{3}+\left(85a-255\right){x}+684a-1938$ |
| 256.1-b4 |
256.1-b |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.665556220$ |
$17.30576610$ |
2.513424990 |
\( 16581375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -85 a - 170\) , \( 684 a + 1254\bigr] \) |
${y}^2={x}^{3}+\left(-85a-170\right){x}+684a+1254$ |
| 256.1-c1 |
256.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.856120083$ |
1.059692279 |
\( 10220 a - 27944 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a + 11\) , \( 17 a + 30\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(5a+11\right){x}+17a+30$ |
| 256.1-c2 |
256.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.856120083$ |
1.059692279 |
\( -481853450 a + 1344996078 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -35 a - 69\) , \( 121 a + 214\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-35a-69\right){x}+121a+214$ |
| 256.1-d1 |
256.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 7$ |
2B, 7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}$ |
| 256.1-d2 |
256.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 7$ |
2B, 7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 2\) , \( 2\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+2$ |
| 256.1-d3 |
256.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 54000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 8\) , \( 16 a + 28\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-8\right){x}+16a+28$ |
| 256.1-d4 |
256.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 54000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a - 13\) , \( -11 a + 31\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-13\right){x}-11a+31$ |
| 256.1-e1 |
256.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.561416932$ |
$12.75994948$ |
3.126473920 |
\( 10220 a - 27944 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 5 a + 11\) , \( -17 a - 30\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(5a+11\right){x}-17a-30$ |
| 256.1-e2 |
256.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.122833865$ |
$12.75994948$ |
3.126473920 |
\( -481853450 a + 1344996078 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -35 a - 69\) , \( -121 a - 214\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-35a-69\right){x}-121a-214$ |
| 256.1-f1 |
256.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.787454598$ |
$4.856120083$ |
1.668919117 |
\( -10220 a - 17724 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3 a - 4\) , \( -5 a - 8\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-3a-4\right){x}-5a-8$ |
| 256.1-f2 |
256.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.574909197$ |
$4.856120083$ |
1.668919117 |
\( 481853450 a + 863142628 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -43 a - 84\) , \( -261 a - 464\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-43a-84\right){x}-261a-464$ |
| 256.1-g1 |
256.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.75994948$ |
2.784449256 |
\( 10220 a - 27944 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 3 a - 7\) , \( -5 a + 13\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(3a-7\right){x}-5a+13$ |
| 256.1-g2 |
256.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.75994948$ |
2.784449256 |
\( -481853450 a + 1344996078 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 43 a - 127\) , \( -261 a + 725\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(43a-127\right){x}-261a+725$ |
| 256.1-h1 |
256.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.856120083$ |
1.059692279 |
\( -10220 a - 17724 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -5 a + 16\) , \( -17 a + 47\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-5a+16\right){x}-17a+47$ |
| 256.1-h2 |
256.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.856120083$ |
1.059692279 |
\( 481853450 a + 863142628 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 35 a - 104\) , \( -121 a + 335\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(35a-104\right){x}-121a+335$ |
| 256.1-i1 |
256.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.561416932$ |
$12.75994948$ |
3.126473920 |
\( -10220 a - 17724 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -5 a + 16\) , \( 17 a - 47\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-5a+16\right){x}+17a-47$ |
| 256.1-i2 |
256.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.122833865$ |
$12.75994948$ |
3.126473920 |
\( 481853450 a + 863142628 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 35 a - 104\) , \( 121 a - 335\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(35a-104\right){x}+121a-335$ |
| 256.1-j1 |
256.1-j |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.332778110$ |
$17.30576610$ |
2.513424990 |
\( -3375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a - 15\) , \( -12 a + 34\bigr] \) |
${y}^2={x}^{3}+\left(5a-15\right){x}-12a+34$ |
| 256.1-j2 |
256.1-j |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$2.329446770$ |
$2.472252300$ |
2.513424990 |
\( -3375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -5 a - 10\) , \( -12 a - 22\bigr] \) |
${y}^2={x}^{3}+\left(-5a-10\right){x}-12a-22$ |
| 256.1-j3 |
256.1-j |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.665556220$ |
$17.30576610$ |
2.513424990 |
\( 16581375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 85 a - 255\) , \( -684 a + 1938\bigr] \) |
${y}^2={x}^{3}+\left(85a-255\right){x}-684a+1938$ |
| 256.1-j4 |
256.1-j |
$4$ |
$14$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$4.658893541$ |
$2.472252300$ |
2.513424990 |
\( 16581375 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -85 a - 170\) , \( -684 a - 1254\bigr] \) |
${y}^2={x}^{3}+\left(-85a-170\right){x}-684a-1254$ |
| 256.1-k1 |
256.1-k |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.787454598$ |
$4.856120083$ |
1.668919117 |
\( 10220 a - 27944 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 3 a - 7\) , \( 5 a - 13\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(3a-7\right){x}+5a-13$ |
| 256.1-k2 |
256.1-k |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{22} \) |
$1.63798$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.574909197$ |
$4.856120083$ |
1.668919117 |
\( -481853450 a + 1344996078 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 43 a - 127\) , \( 261 a - 725\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(43a-127\right){x}+261a-725$ |
*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.
