| 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 |
| 38416.1-a1 |
38416.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 3 \) |
$0.732377379$ |
$0.528778102$ |
3.286053516 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -343\) , \( 2401\bigr] \) |
${y}^2={x}^{3}-343{x}+2401$ |
| 38416.1-b1 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{48} \cdot 7^{14} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$3.507207167$ |
$0.062529795$ |
1.240576118 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -33421\) , \( 2364150\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-33421{x}+2364150$ |
| 38416.1-b2 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{16} \cdot 7^{14} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.389689685$ |
$0.562768158$ |
1.240576118 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -101\) , \( -786\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-101{x}-786$ |
| 38416.1-b3 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{24} \cdot 7^{18} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$1.169069055$ |
$0.187589386$ |
1.240576118 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 879\) , \( 16658\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+879{x}+16658$ |
| 38416.1-b4 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{18} \cdot 7^{24} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$2.338138111$ |
$0.093794693$ |
1.240576118 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -6961\) , \( 184434\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-6961{x}+184434$ |
| 38416.1-b5 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{14} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.779379370$ |
$0.281384079$ |
1.240576118 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -2061\) , \( -35674\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-2061{x}-35674$ |
| 38416.1-b6 |
38416.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{30} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$7.014414334$ |
$0.031264897$ |
1.240576118 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -535181\) , \( 150784758\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-535181{x}+150784758$ |
| 38416.1-c1 |
38416.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{20} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$4$ |
\( 1 \) |
$1$ |
$0.303563928$ |
0.858608448 |
\( 12544 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -800\) , \( 8359\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-800{x}+8359$ |
| 38416.1-d1 |
38416.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.699215416$ |
$0.645823970$ |
3.831699161 |
\( 1792 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -114\) , \( 127\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-114{x}+127$ |
| 38416.1-d2 |
38416.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \cdot 3 \) |
$2.097646250$ |
$0.215274656$ |
3.831699161 |
\( 406749952 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -6974\) , \( 226507\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-6974{x}+226507$ |
| 38416.1-e1 |
38416.1-e |
$4$ |
$14$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{12} \cdot 7^{6} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.431581047$ |
$2.472252300$ |
3.017867362 |
\( -3375 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -8\) , \( -8\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-8{x}-8$ |
| 38416.1-e2 |
38416.1-e |
$4$ |
$14$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{12} \cdot 7^{18} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$3.021067335$ |
$0.353178900$ |
3.017867362 |
\( -3375 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -428\) , \( 4416\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-428{x}+4416$ |
| 38416.1-e3 |
38416.1-e |
$4$ |
$14$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{12} \cdot 7^{18} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$6.042134670$ |
$0.176589450$ |
3.017867362 |
\( 16581375 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -7288\) , \( 243144\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-7288{x}+243144$ |
| 38416.1-e4 |
38416.1-e |
$4$ |
$14$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{12} \cdot 7^{6} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.863162095$ |
$1.236126150$ |
3.017867362 |
\( 16581375 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -148\) , \( -624\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-148{x}-624$ |
| 38416.1-f1 |
38416.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{4} \cdot 7^{14} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.147513062$ |
3.245657071 |
\( \frac{432}{7} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 13\) , \( -92\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+13{x}-92$ |
| 38416.1-f2 |
38416.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{10} \cdot 7^{20} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.286878265$ |
3.245657071 |
\( \frac{11090466}{2401} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -722\) , \( 6278\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-722{x}+6278$ |
| 38416.1-f3 |
38416.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$0.573756531$ |
3.245657071 |
\( \frac{740772}{49} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -232\) , \( -1170\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-232{x}-1170$ |
| 38416.1-f4 |
38416.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{10} \cdot 7^{14} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$16$ |
\( 2^{4} \) |
$1$ |
$0.286878265$ |
3.245657071 |
\( \frac{1443468546}{7} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -3662\) , \( -83490\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-3662{x}-83490$ |
| 38416.1-g1 |
38416.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{8} \) |
$3.53844$ |
$(a), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$1$ |
\( 3 \) |
$0.538281715$ |
$2.124947498$ |
9.705637806 |
\( 12544 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -16\) , \( -29\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-16{x}-29$ |
| 38416.1-h1 |
38416.1-h |
$2$ |
$3$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{4} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \) |
$0.552001852$ |
$4.520767792$ |
7.058261248 |
\( 1792 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -2\) , \( -1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-2{x}-1$ |
| 38416.1-h2 |
38416.1-h |
$2$ |
$3$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{4} \) |
$3.53844$ |
$(a), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 2 \) |
$1.656005557$ |
$1.506922597$ |
7.058261248 |
\( 406749952 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -142\) , \( -701\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-142{x}-701$ |
| 38416.1-i1 |
38416.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{14} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.913677040$ |
5.168537851 |
\( -\frac{4}{7} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -4\) , \( 174\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-4{x}+174$ |
| 38416.1-i2 |
38416.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{10} \cdot 7^{16} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.456838520$ |
5.168537851 |
\( \frac{3543122}{49} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -494\) , \( 4094\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-494{x}+4094$ |
| 38416.1-j1 |
38416.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
38416.1 |
\( 2^{4} \cdot 7^{4} \) |
\( 2^{8} \cdot 7^{4} \) |
$3.53844$ |
$(a), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cn |
$4$ |
\( 1 \) |
$1$ |
$3.701446717$ |
10.46927229 |
\( 48384 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( -7\bigr] \) |
${y}^2={x}^{3}-7{x}-7$ |
*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.
