| 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 |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-24$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$796.3913345$ |
1.036967883 |
\( -1707264 a^{3} + 5121792 a + 2417472 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -5 a^{2} - 12 a - 5\) , \( -10 a^{3} - 16 a^{2} + 12 a + 9\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}-a{x}^{2}+\left(-5a^{2}-12a-5\right){x}-10a^{3}-16a^{2}+12a+9$ |
| 16.1-a2 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-24$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$796.3913345$ |
1.036967883 |
\( 1707264 a^{3} - 5121792 a + 2417472 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{2} - a + 2\) , \( 0\) , \( -5 a^{2} + 10 a - 2\) , \( -10 a^{3} + 11 a^{2} + 23 a - 13\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-5a^{2}+10a-2\right){x}-10a^{3}+11a^{2}+23a-13$ |
| 16.1-a3 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-24$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$796.3913345$ |
1.036967883 |
\( -1707264 a^{3} + 5121792 a + 2417472 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{2} - a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -10 a^{3} + 3 a^{2} + 38 a - 21\) , \( -27 a^{3} + 15 a^{2} + 97 a - 53\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-10a^{3}+3a^{2}+38a-21\right){x}-27a^{3}+15a^{2}+97a-53$ |
| 16.1-a4 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-24$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$796.3913345$ |
1.036967883 |
\( 1707264 a^{3} - 5121792 a + 2417472 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a\) , \( 0\) , \( 10 a^{3} + 5 a^{2} - 40 a - 22\) , \( -17 a^{3} - 11 a^{2} + 58 a + 31\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}={x}^{3}-a{x}^{2}+\left(10a^{3}+5a^{2}-40a-22\right){x}-17a^{3}-11a^{2}+58a+31$ |
| 16.1-a5 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( 6508137062232 a^{3} - 3368859648336 a^{2} - 24288698179512 a + 12572755370856 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 5 a + 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -7 a^{3} + 5 a^{2} + 23 a - 17\) , \( 8 a^{3} - 4 a^{2} - 32 a + 14\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-7a^{3}+5a^{2}+23a-17\right){x}+8a^{3}-4a^{2}-32a+14$ |
| 16.1-a6 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( -1743850069416 a^{3} + 3368859648336 a^{2} + 467263215432 a - 902683222488 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 5 a + 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 4 a^{3} - 5 a^{2} - 10 a + 3\) , \( -a^{3} + 3 a^{2} - 5 a\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(4a^{3}-5a^{2}-10a+3\right){x}-a^{3}+3a^{2}-5a$ |
| 16.1-a7 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( -6508137062232 a^{3} - 3368859648336 a^{2} + 24288698179512 a + 12572755370856 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 5 a + 1\) , \( a^{2} - 1\) , \( 9 a^{3} + 5 a^{2} - 35 a - 16\) , \( 16 a^{3} + 8 a^{2} - 60 a - 31\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(9a^{3}+5a^{2}-35a-16\right){x}+16a^{3}+8a^{2}-60a-31$ |
| 16.1-a8 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( 1743850069416 a^{3} + 3368859648336 a^{2} - 467263215432 a - 902683222488 \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 5 a + 1\) , \( a^{2} - 1\) , \( -2 a^{3} - 5 a^{2} - 2 a + 4\) , \( -4 a^{3} - 9 a^{2} + 3\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-2a^{3}-5a^{2}-2a+4\right){x}-4a^{3}-9a^{2}+3$ |
| 16.1-a9 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( -6508137062232 a^{3} - 3368859648336 a^{2} + 24288698179512 a + 12572755370856 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 5 a + 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 7 a^{3} + 5 a^{2} - 25 a - 17\) , \( -8 a^{3} - 4 a^{2} + 30 a + 14\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(7a^{3}+5a^{2}-25a-17\right){x}-8a^{3}-4a^{2}+30a+14$ |
| 16.1-a10 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( 1743850069416 a^{3} + 3368859648336 a^{2} - 467263215432 a - 902683222488 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 5 a + 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -4 a^{3} - 5 a^{2} + 8 a + 3\) , \( a^{3} + 3 a^{2} + 3 a\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-4a^{3}-5a^{2}+8a+3\right){x}+a^{3}+3a^{2}+3a$ |
| 16.1-a11 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( 6508137062232 a^{3} - 3368859648336 a^{2} - 24288698179512 a + 12572755370856 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 5 a + 1\) , \( a^{2} - 1\) , \( -9 a^{3} + 5 a^{2} + 33 a - 16\) , \( -16 a^{3} + 8 a^{2} + 60 a - 31\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-9a^{3}+5a^{2}+33a-16\right){x}-16a^{3}+8a^{2}+60a-31$ |
| 16.1-a12 |
16.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-96$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$199.0978336$ |
1.036967883 |
\( -1743850069416 a^{3} + 3368859648336 a^{2} + 467263215432 a - 902683222488 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 5 a + 1\) , \( a^{2} - 1\) , \( 2 a^{3} - 5 a^{2} + 4\) , \( 4 a^{3} - 9 a^{2} + 3\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(2a^{3}-5a^{2}+4\right){x}+4a^{3}-9a^{2}+3$ |
| 16.1-b1 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( -600840130180059000 a^{3} + 1160733998424384000 a^{2} + 160994627660022750 a - 311017737504159000 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{2} - 1\) , \( a^{3} - 4 a + 1\) , \( -49 a^{3} - 26 a^{2} + 177 a + 85\) , \( -168 a^{3} - 86 a^{2} + 634 a + 334\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(-49a^{3}-26a^{2}+177a+85\right){x}-168a^{3}-86a^{2}+634a+334$ |
| 16.1-b2 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$2504.913233$ |
0.815401443 |
\( 818626500 a^{2} - 219348000 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{2} - 1\) , \( a^{3} - 4 a + 1\) , \( 11 a^{3} + 4 a^{2} - 48 a - 25\) , \( -5 a^{3} + a^{2} + 25 a + 10\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(11a^{3}+4a^{2}-48a-25\right){x}-5a^{3}+a^{2}+25a+10$ |
| 16.1-b3 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( 600840130180059000 a^{3} + 1160733998424384000 a^{2} - 160994627660022750 a - 311017737504159000 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 49 a^{3} - 27 a^{2} - 175 a + 89\) , \( 81 a^{3} - 38 a^{2} - 313 a + 160\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(49a^{3}-27a^{2}-175a+89\right){x}+81a^{3}-38a^{2}-313a+160$ |
| 16.1-b4 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 3\) , \( a + 1\) , \( 19 a^{3} + 25 a^{2} - 28 a - 17\) , \( 38 a^{3} + 86 a^{2} + 15 a - 10\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(19a^{3}+25a^{2}-28a-17\right){x}+38a^{3}+86a^{2}+15a-10$ |
| 16.1-b5 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( -2242365893060213250 a^{3} - 1160733998424384000 a^{2} + 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} - a^{2} + 5 a + 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -21 a^{3} + 26 a^{2} + 34 a - 17\) , \( -12 a^{3} + 37 a^{2} - 34 a + 10\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+3\right){x}^{2}+\left(-21a^{3}+26a^{2}+34a-17\right){x}-12a^{3}+37a^{2}-34a+10$ |
| 16.1-b6 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$2504.913233$ |
0.815401443 |
\( -818626500 a^{2} + 3055158000 \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 3\) , \( a + 1\) , \( 4 a^{3} - 5 a^{2} - 28 a - 7\) , \( -5 a^{3} - a^{2} + 24 a + 14\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{3}-5a^{2}-28a-7\right){x}-5a^{3}-a^{2}+24a+14$ |
| 16.1-b7 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1252.456616$ |
0.815401443 |
\( 54000 \) |
\( \bigl[a^{2} - 1\) , \( -1\) , \( a^{2} - 1\) , \( -a^{2} - 1\) , \( -a^{2} + 1\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}-{x}^{2}+\left(-a^{2}-1\right){x}-a^{2}+1$ |
| 16.1-b8 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1252.456616$ |
0.815401443 |
\( 54000 \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} + 1\) , \( a^{2} - 1\) , \( -a^{2} - 1\) , \( -1\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-a^{2}-1\right){x}-1$ |
| 16.1-b9 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + 5 a\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 2\) , \( 2 a^{3} - 2 a^{2} - 3 a - 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+2{x}+2a^{3}-2a^{2}-3a-1$ |
| 16.1-b10 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$626.2283084$ |
0.815401443 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + 5 a\) , \( a + 1\) , \( 2\) , \( a^{3} + 2 a^{2} - 1\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+2{x}+a^{3}+2a^{2}-1$ |
| 16.1-b11 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$16$ |
\( 1 \) |
$1$ |
$9.784817320$ |
0.815401443 |
\( -2242365893060213250 a^{3} - 1160733998424384000 a^{2} + 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} - 4 a + 1\) , \( a^{2} + a - 2\) , \( a^{2} - 1\) , \( -18 a^{3} + 27 a^{2} + 25 a - 22\) , \( 19 a^{3} - 60 a^{2} + 43 a - 11\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-18a^{3}+27a^{2}+25a-22\right){x}+19a^{3}-60a^{2}+43a-11$ |
| 16.1-b12 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 1 \) |
$1$ |
$156.5570771$ |
0.815401443 |
\( -818626500 a^{2} + 3055158000 \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 1\) , \( 4 a^{3} - 3 a^{2} - 28 a - 12\) , \( 9 a^{3} - 3 a^{2} - 52 a - 25\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(4a^{3}-3a^{2}-28a-12\right){x}+9a^{3}-3a^{2}-52a-25$ |
| 16.1-b13 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$16$ |
\( 1 \) |
$1$ |
$9.784817320$ |
0.815401443 |
\( 2242365893060213250 a^{3} - 1160733998424384000 a^{2} - 8368623442060794000 a + 4331918256193377000 \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 1\) , \( 19 a^{3} + 27 a^{2} - 28 a - 22\) , \( -19 a^{3} - 60 a^{2} - 43 a - 11\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(19a^{3}+27a^{2}-28a-22\right){x}-19a^{3}-60a^{2}-43a-11$ |
| 16.1-b14 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$16$ |
\( 1 \) |
$1$ |
$9.784817320$ |
0.815401443 |
\( -600840130180059000 a^{3} + 1160733998424384000 a^{2} + 160994627660022750 a - 311017737504159000 \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} - 1\) , \( -48 a^{3} - 28 a^{2} + 172 a + 88\) , \( 119 a^{3} + 59 a^{2} - 457 a - 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-48a^{3}-28a^{2}+172a+88\right){x}+119a^{3}+59a^{2}-457a-249$ |
| 16.1-b15 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 1 \) |
$1$ |
$156.5570771$ |
0.815401443 |
\( 818626500 a^{2} - 219348000 \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} - 1\) , \( 12 a^{3} + 2 a^{2} - 53 a - 22\) , \( 16 a^{3} + 2 a^{2} - 73 a - 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(12a^{3}+2a^{2}-53a-22\right){x}+16a^{3}+2a^{2}-73a-35$ |
| 16.1-b16 |
16.1-b |
$16$ |
$48$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.06589$ |
$(a^3-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-192$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$16$ |
\( 1 \) |
$1$ |
$9.784817320$ |
0.815401443 |
\( 600840130180059000 a^{3} + 1160733998424384000 a^{2} - 160994627660022750 a - 311017737504159000 \) |
\( \bigl[a + 1\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{2} - 1\) , \( 47 a^{3} - 28 a^{2} - 171 a + 88\) , \( -119 a^{3} + 59 a^{2} + 457 a - 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(47a^{3}-28a^{2}-171a+88\right){x}-119a^{3}+59a^{2}+457a-249$ |
*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.
