| 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 |
| 405.1-a1 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{44} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.186308476$ |
1.885309221 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -3955\) , \( -178157\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-3955{x}-178157$ |
| 405.1-a2 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{14} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$2.980935616$ |
1.885309221 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5\) , \( 43\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+5{x}+43$ |
| 405.1-a3 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{16} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.372616952$ |
1.885309221 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 1265\) , \( -9785\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+1265{x}-9785$ |
| 405.1-a4 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{20} \cdot 5^{8} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$0.745233904$ |
1.885309221 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -355\) , \( -1037\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-355{x}-1037$ |
| 405.1-a5 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{4} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.490467808$ |
1.885309221 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -175\) , \( 1015\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-175{x}+1015$ |
| 405.1-a6 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{28} \cdot 5^{4} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$16$ |
\( 2^{4} \) |
$1$ |
$0.372616952$ |
1.885309221 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -4855\) , \( -127937\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-4855{x}-127937$ |
| 405.1-a7 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{14} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.745233904$ |
1.885309221 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -2875\) , \( 60955\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-2875{x}+60955$ |
| 405.1-a8 |
405.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 2^{12} \cdot 3^{20} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.186308476$ |
1.885309221 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -77755\) , \( -8307317\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-77755{x}-8307317$ |
| 405.1-b1 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{44} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$5.036899059$ |
$0.186308476$ |
4.748056122 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -990\) , \( 22765\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-990{x}+22765$ |
| 405.1-b2 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.259224764$ |
$2.980935616$ |
4.748056122 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 0\) , \( -5\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-5$ |
| 405.1-b3 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$10.07379811$ |
$0.372616952$ |
4.748056122 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 315\) , \( 1066\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2+315{x}+1066$ |
| 405.1-b4 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{20} \cdot 5^{8} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$5.036899059$ |
$0.745233904$ |
4.748056122 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -90\) , \( 175\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-90{x}+175$ |
| 405.1-b5 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$2.518449529$ |
$1.490467808$ |
4.748056122 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -45\) , \( -104\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-45{x}-104$ |
| 405.1-b6 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{28} \cdot 5^{4} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$10.07379811$ |
$0.372616952$ |
4.748056122 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1215\) , \( 16600\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-1215{x}+16600$ |
| 405.1-b7 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$5.036899059$ |
$0.745233904$ |
4.748056122 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -720\) , \( -7259\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-720{x}-7259$ |
| 405.1-b8 |
405.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{20} \cdot 5^{2} \) |
$2.53532$ |
$(5,a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$20.14759623$ |
$0.186308476$ |
4.748056122 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -19440\) , \( 1048135\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-19440{x}+1048135$ |
*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.
