| 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 |
| 20.1-a1 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.772687765$ |
2.059677057 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -71 a - 272\) , \( -1174 a - 4546\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-71a-272\right){x}-1174a-4546$ |
| 20.1-a2 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.95418988$ |
2.059677057 |
\( \frac{21296}{25} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 9 a + 38\) , \( 40 a + 156\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(9a+38\right){x}+40a+156$ |
| 20.1-a3 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$31.90837977$ |
2.059677057 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -42 a - 160\) , \( 274 a + 1062\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-42a-160\right){x}+274a+1062$ |
| 20.1-a4 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.545375530$ |
2.059677057 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1322 a - 5120\) , \( -49390 a - 191290\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1322a-5120\right){x}-49390a-191290$ |
| 20.1-b1 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$10.34365470$ |
1.335360080 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -75 a - 283\) , \( 956 a + 3708\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-75a-283\right){x}+956a+3708$ |
| 20.1-b2 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$10.34365470$ |
1.335360080 |
\( \frac{21296}{25} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 5 a + 27\) , \( -18 a - 64\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(5a+27\right){x}-18a-64$ |
| 20.1-b3 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$20.68730941$ |
1.335360080 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -42 a - 160\) , \( -274 a - 1062\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-42a-160\right){x}-274a-1062$ |
| 20.1-b4 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$20.68730941$ |
1.335360080 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -1322 a - 5120\) , \( 49390 a + 191290\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-1322a-5120\right){x}+49390a+191290$ |
| 20.1-c1 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{12} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.957327600$ |
$10.34365470$ |
2.613737142 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -36\) , \( 140\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-36{x}+140$ |
| 20.1-c2 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.652442533$ |
$10.34365470$ |
2.613737142 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 4\) , \( -4\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+4{x}-4$ |
| 20.1-c3 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.326221266$ |
$20.68730941$ |
2.613737142 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-{x}$ |
| 20.1-c4 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.978663800$ |
$20.68730941$ |
2.613737142 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -41\) , \( 116\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-41{x}+116$ |
| 20.1-d1 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{12} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$7.569157173$ |
$1.772687765$ |
1.732224375 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -36\) , \( -140\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-36{x}-140$ |
| 20.1-d2 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.523052391$ |
$15.95418988$ |
1.732224375 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 4\) , \( 4\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+4{x}+4$ |
| 20.1-d3 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.261526195$ |
$31.90837977$ |
1.732224375 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-{x}$ |
| 20.1-d4 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.46377$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$3.784578586$ |
$3.545375530$ |
1.732224375 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -41\) , \( -116\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-41{x}-116$ |
*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.
