| 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 |
| 1.1-a3 |
1.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.43777$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.639994193$ |
0.287814748 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 67 a - 161\) , \( 458 a - 1122\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(67a-161\right){x}+458a-1122$ |
| 1.1-a4 |
1.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.43777$ |
$\textsf{none}$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$50.75994773$ |
0.287814748 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -18 a - 41\) , \( 56 a + 138\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-18a-41\right){x}+56a+138$ |
| 16.1-a3 |
16.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$0.87554$ |
$(-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$2.819997096$ |
1.295166367 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 264 a - 646\) , \( 3782 a - 9264\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(264a-646\right){x}+3782a-9264$ |
| 16.1-a4 |
16.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$0.87554$ |
$(-a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$25.37997386$ |
1.295166367 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -64 a - 166\) , \( 418 a + 1056\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-64a-166\right){x}+418a+1056$ |
| 225.2-e3 |
225.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.2 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.792599469$ |
$3.089152043$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -1029 a - 2526\) , \( 28451 a + 69661\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-1029a-2526\right){x}+28451a+69661$ |
| 225.2-e4 |
225.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.2 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.896299734$ |
$6.178304087$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 226 a - 606\) , \( -2841 a + 6833\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(226a-606\right){x}-2841a+6833$ |
| 225.3-e3 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.792599469$ |
$3.089152043$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 4118 a - 10086\) , \( 223834 a - 548279\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(4118a-10086\right){x}+223834a-548279$ |
| 225.3-e4 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.896299734$ |
$6.178304087$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -17 a - 246\) , \( -714 a - 547\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-17a-246\right){x}-714a-547$ |
| 529.2-e3 |
529.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
529.2 |
\( 23^{2} \) |
\( 23^{6} \) |
$2.09946$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$3.756467216$ |
$2.494715329$ |
3.825823879 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -228 a - 675\) , \( 3440 a + 7933\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-228a-675\right){x}+3440a+7933$ |
| 529.2-e4 |
529.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
529.2 |
\( 23^{2} \) |
\( 23^{6} \) |
$2.09946$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.878233608$ |
$4.989430659$ |
3.825823879 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 2294 a - 5633\) , \( -95269 a + 233329\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2294a-5633\right){x}-95269a+233329$ |
| 529.3-e3 |
529.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
529.3 |
\( 23^{2} \) |
\( 23^{6} \) |
$2.09946$ |
$(2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.878233608$ |
$4.989430659$ |
3.825823879 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -565 a - 1433\) , \( -11357 a - 27731\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-565a-1433\right){x}-11357a-27731$ |
| 529.3-e4 |
529.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
529.3 |
\( 23^{2} \) |
\( 23^{6} \) |
$2.09946$ |
$(2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$3.756467216$ |
$2.494715329$ |
3.825823879 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 997 a - 2475\) , \( 27254 a - 66707\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(997a-2475\right){x}+27254a-66707$ |
| 625.1-b3 |
625.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.18884$ |
$(-a-1), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 2^{2} \) |
$1$ |
$1.127998838$ |
2.072266188 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 1645 a - 4053\) , \( 58718 a - 143855\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1645a-4053\right){x}+58718a-143855$ |
| 625.1-b4 |
625.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.18884$ |
$(-a-1), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$10.15198954$ |
2.072266188 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -396 a - 1053\) , \( 6656 a + 16645\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-396a-1053\right){x}+6656a+16645$ |
*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.
