| 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 |
| 72.2-a5 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{20} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.908836754$ |
0.642644632 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 70 a + 85\) , \( -98 a + 559\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(70a+85\right){x}-98a+559$ |
| 144.2-a5 |
144.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
144.2 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{11} \cdot 3^{20} \) |
$0.87554$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \) |
$1$ |
$0.908836754$ |
1.285289264 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 70 a + 85\) , \( 98 a - 558\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(70a+85\right){x}+98a-558$ |
| 648.3-a5 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{32} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.158547729$ |
$0.302945584$ |
1.849572148 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 630 a + 757\) , \( 2646 a - 15079\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(630a+757\right){x}+2646a-15079$ |
| 1296.3-b5 |
1296.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1296.3 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{32} \) |
$1.51647$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.302945584$ |
1.713719018 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 630 a + 756\) , \( -3276 a + 14324\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(630a+756\right){x}-3276a+14324$ |
| 6912.2-a5 |
6912.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{23} \cdot 3^{26} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.929871924$ |
$0.262358572$ |
2.760090861 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 392 a - 1456\) , \( 8388 a - 20772\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(392a-1456\right){x}+8388a-20772$ |
| 6912.2-n5 |
6912.2-n |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{23} \cdot 3^{26} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.262358572$ |
2.968248410 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 392 a - 1456\) , \( -8388 a + 20772\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(392a-1456\right){x}-8388a+20772$ |
| 6912.3-a5 |
6912.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{23} \cdot 3^{26} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.719487697$ |
$0.262358572$ |
2.760090861 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -952 a + 784\) , \( -548 a - 23908\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-952a+784\right){x}-548a-23908$ |
| 6912.3-n5 |
6912.3-n |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{23} \cdot 3^{26} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.262358572$ |
2.968248410 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -952 a + 784\) , \( 548 a + 23908\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-952a+784\right){x}+548a+23908$ |
| 9216.2-m5 |
9216.2-m |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{29} \cdot 3^{20} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.321322316$ |
1.817673508 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -560 a - 672\) , \( -8936 a - 3136\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-560a-672\right){x}-8936a-3136$ |
| 9216.2-o5 |
9216.2-o |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{29} \cdot 3^{20} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \) |
$1$ |
$0.321322316$ |
1.817673508 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -560 a - 672\) , \( 8936 a + 3136\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-560a-672\right){x}+8936a+3136$ |
| 20808.4-c5 |
20808.4-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20808.4 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{11} \cdot 3^{20} \cdot 17^{6} \) |
$3.03558$ |
$(a), (-a-1), (a-1), (-2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.220425290$ |
2.493827480 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -936 a + 1764\) , \( -15876 a - 34344\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-936a+1764\right){x}-15876a-34344$ |
| 20808.6-c5 |
20808.6-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20808.6 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{11} \cdot 3^{20} \cdot 17^{6} \) |
$3.03558$ |
$(a), (-a-1), (a-1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.220425290$ |
2.493827480 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1076 a - 1596\) , \( 24556 a - 16088\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1076a-1596\right){x}+24556a-16088$ |
| 26136.4-d5 |
26136.4-d |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.4 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{11} \cdot 3^{26} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1.533166001$ |
$0.158208171$ |
5.488492471 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -1499 a - 3721\) , \( -51754 a - 78790\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1499a-3721\right){x}-51754a-78790$ |
| 26136.6-i5 |
26136.6-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.6 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{11} \cdot 3^{26} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$0.158208171$ |
3.579842277 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2868 a - 1374\) , \( -71480 a - 30430\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2868a-1374\right){x}-71480a-30430$ |
| 26136.7-i5 |
26136.7-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.7 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{11} \cdot 3^{26} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \) |
$1$ |
$0.158208171$ |
3.579842277 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -488 a + 4226\) , \( 74840 a + 27698\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-488a+4226\right){x}+74840a+27698$ |
| 26136.9-d5 |
26136.9-d |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.9 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{11} \cdot 3^{26} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1.533166001$ |
$0.158208171$ |
5.488492471 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -2841 a - 1481\) , \( 72614 a - 24638\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2841a-1481\right){x}+72614a-24638$ |
| 27648.2-v5 |
27648.2-v |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{29} \cdot 3^{26} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$2.015252848$ |
$0.185515525$ |
4.229750882 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -782 a + 2911\) , \( 40761 a + 36463\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-782a+2911\right){x}+40761a+36463$ |
| 27648.2-bb5 |
27648.2-bb |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{29} \cdot 3^{26} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.185515525$ |
2.098868579 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -782 a + 2911\) , \( -40761 a - 36463\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-782a+2911\right){x}-40761a-36463$ |
| 27648.3-f5 |
27648.3-f |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{29} \cdot 3^{26} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.061011393$ |
$0.185515525$ |
4.229750882 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 1902 a - 1569\) , \( -45913 a + 623\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(1902a-1569\right){x}-45913a+623$ |
| 27648.3-bp5 |
27648.3-bp |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{29} \cdot 3^{26} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.185515525$ |
2.098868579 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 1902 a - 1569\) , \( 45913 a - 623\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(1902a-1569\right){x}+45913a-623$ |
| 41616.4-c5 |
41616.4-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
41616.4 |
\( 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{11} \cdot 3^{20} \cdot 17^{6} \) |
$3.60993$ |
$(a), (-a-1), (a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1.036050787$ |
$0.220425290$ |
5.167463847 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -938 a + 1765\) , \( 16813 a + 32581\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-938a+1765\right){x}+16813a+32581$ |
| 41616.6-g5 |
41616.6-g |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
41616.6 |
\( 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{11} \cdot 3^{20} \cdot 17^{6} \) |
$3.60993$ |
$(a), (-a-1), (a-1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{8} \) |
$0.259012696$ |
$0.220425290$ |
5.167463847 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 1078 a - 1595\) , \( -25633 a + 17685\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1078a-1595\right){x}-25633a+17685$ |
| 45000.2-i5 |
45000.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
45000.2 |
\( 2^{3} \cdot 3^{2} \cdot 5^{4} \) |
\( 2^{11} \cdot 3^{20} \cdot 5^{12} \) |
$3.68118$ |
$(a), (-a-1), (a-1), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \) |
$1.068541721$ |
$0.181767350$ |
8.789665301 |
\( -\frac{2327042746553}{43046721} a + \frac{2081263802600}{43046721} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 1750 a + 2098\) , \( -14000 a + 67714\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(1750a+2098\right){x}-14000a+67714$ |
*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.
