| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Ns, 5B |
$1$ |
\( 3 \cdot 5 \) |
$0.189214982$ |
$13.64906546$ |
3.990331883 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -595 a - 5443\) , \( 26767 a + 246519\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-595a-5443\right){x}+26767a+246519$ |
| 4.1-a2 |
4.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Ns, 5B |
$1$ |
\( 3 \) |
$0.946074912$ |
$13.64906546$ |
3.990331883 |
\( \frac{1331}{8} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 5 a + 82\) , \( -87 a - 759\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(5a+82\right){x}-87a-759$ |
| 4.1-b1 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{20} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$1.429861641$ |
$3.921006551$ |
4.619988463 |
\( -\frac{1030301}{16} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 18636 a - 190049\) , \( 4508944 a - 46027693\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(18636a-190049\right){x}+4508944a-46027693$ |
| 4.1-b2 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{52} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.285972328$ |
$3.921006551$ |
4.619988463 |
\( \frac{237176659}{1048576} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -114189 a + 1165861\) , \( -169485476 a + 1730149787\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-114189a+1165861\right){x}-169485476a+1730149787$ |
| 4.1-c1 |
4.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{20} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$1.429861641$ |
$3.921006551$ |
4.619988463 |
\( -\frac{1030301}{16} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 18 a + 212\) , \( 56 a + 572\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(18a+212\right){x}+56a+572$ |
| 4.1-c2 |
4.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{52} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.285972328$ |
$3.921006551$ |
4.619988463 |
\( \frac{237176659}{1048576} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 33 a + 362\) , \( 461 a + 4382\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(33a+362\right){x}+461a+4382$ |
| 4.1-d1 |
4.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{30} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Ns, 5B |
$1$ |
\( 3 \cdot 5 \) |
$0.189214982$ |
$13.64906546$ |
3.990331883 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 595 a - 6038\) , \( -26767 a + 273286\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(595a-6038\right){x}-26767a+273286$ |
| 4.1-d2 |
4.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$2.45372$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Ns, 5B |
$1$ |
\( 3 \) |
$0.946074912$ |
$13.64906546$ |
3.990331883 |
\( \frac{1331}{8} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -5 a + 87\) , \( 87 a - 846\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-5a+87\right){x}+87a-846$ |
| 8.3-a1 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{10} \) |
$2.91798$ |
$(2,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
|
\( 2 \) |
$1$ |
$4.710212463$ |
5.109411999 |
\( -508805125 a + 5194007250 \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 5 a + 354\) , \( -158 a - 416\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(5a+354\right){x}-158a-416$ |
| 8.3-a2 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{8} \) |
$2.91798$ |
$(2,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
|
\( 2^{2} \) |
$1$ |
$18.84084985$ |
5.109411999 |
\( -4875 a + 55250 \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -20 a + 124\) , \( -125 a - 110\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-20a+124\right){x}-125a-110$ |
| 8.3-a3 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{22} \) |
$2.91798$ |
$(2,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
|
\( 2 \) |
$1$ |
$18.84084985$ |
5.109411999 |
\( 625 a + 5750 \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 22 a + 186\) , \( 162 a + 235\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22a+186\right){x}+162a+235$ |
| 8.3-a4 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{16} \) |
$2.91798$ |
$(2,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
|
\( 2 \) |
$1$ |
$18.84084985$ |
5.109411999 |
\( 1136625 a + 10468250 \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -140 a - 981\) , \( -3193 a - 28361\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-140a-981\right){x}-3193a-28361$ |
| 8.3-b1 |
8.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{10} \) |
$2.91798$ |
$(2,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$4.710212463$ |
0.485176567 |
\( -508805125 a + 5194007250 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2447 a - 24885\) , \( 232511 a - 2373352\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2447a-24885\right){x}+232511a-2373352$ |
| 8.3-b2 |
8.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{8} \) |
$2.91798$ |
$(2,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( -4875 a + 55250 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 142 a - 1355\) , \( 4778 a - 48598\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(142a-1355\right){x}+4778a-48598$ |
| 8.3-b3 |
8.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{22} \) |
$2.91798$ |
$(2,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( 625 a + 5750 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 340328 a - 3473925\) , \( 6178072729 a - 63067272715\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(340328a-3473925\right){x}+6178072729a-63067272715$ |
| 8.3-b4 |
8.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{16} \) |
$2.91798$ |
$(2,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( 1136625 a + 10468250 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 262 a - 2580\) , \( 331 a - 3202\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(262a-2580\right){x}+331a-3202$ |
| 8.4-a1 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{22} \) |
$2.91798$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$5.265518100$ |
$18.84084985$ |
5.109411999 |
\( -625 a + 6375 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 40 a - 408\) , \( -384 a + 3920\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(40a-408\right){x}-384a+3920$ |
| 8.4-a2 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{16} \) |
$2.91798$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$5.265518100$ |
$18.84084985$ |
5.109411999 |
\( -1136625 a + 11604875 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 1092766 a - 11155180\) , \( 1940333321 a - 19807395878\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(1092766a-11155180\right){x}+1940333321a-19807395878$ |
| 8.4-a3 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{8} \) |
$2.91798$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$10.53103620$ |
$18.84084985$ |
5.109411999 |
\( 4875 a + 50375 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 20 a + 104\) , \( 125 a - 235\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(20a+104\right){x}+125a-235$ |
| 8.4-a4 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{10} \) |
$2.91798$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$21.06207240$ |
$4.710212463$ |
5.109411999 |
\( 508805125 a + 4685202125 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -5 a + 359\) , \( 158 a - 574\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-5a+359\right){x}+158a-574$ |
| 8.4-b1 |
8.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{22} \) |
$2.91798$ |
$(2,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( -625 a + 6375 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -40 a - 368\) , \( -7856 a - 72340\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-40a-368\right){x}-7856a-72340$ |
| 8.4-b2 |
8.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{16} \) |
$2.91798$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( -1136625 a + 11604875 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a - 20\) , \( 9 a - 42\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-20\right){x}+9a-42$ |
| 8.4-b3 |
8.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{8} \) |
$2.91798$ |
$(2,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$18.84084985$ |
0.485176567 |
\( 4875 a + 50375 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -142 a - 1213\) , \( -4778 a - 43820\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-142a-1213\right){x}-4778a-43820$ |
| 8.4-b4 |
8.4-b |
$4$ |
$4$ |
\(\Q(\sqrt{377}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{10} \) |
$2.91798$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$4.710212463$ |
0.485176567 |
\( 508805125 a + 4685202125 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -2447 a - 22438\) , \( -232511 a - 2140841\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-2447a-22438\right){x}-232511a-2140841$ |
*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.
