| 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 |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$2.07008$ |
$(2,a), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$10.34365470$ |
3.776968672 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 404 a - 2180\) , \( -15305 a + 83885\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(404a-2180\right){x}-15305a+83885$ |
| 20.1-a2 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$2.07008$ |
$(2,a), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$10.34365470$ |
3.776968672 |
\( \frac{21296}{25} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -36 a + 230\) , \( 323 a - 1713\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-36a+230\right){x}+323a-1713$ |
| 20.1-a3 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$2.07008$ |
$(2,a), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$20.68730941$ |
3.776968672 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 234 a - 1275\) , \( 2643 a - 14473\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(234a-1275\right){x}+2643a-14473$ |
| 20.1-a4 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$2.07008$ |
$(2,a), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$20.68730941$ |
3.776968672 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 7274 a - 39835\) , \( -805965 a + 4414455\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(7274a-39835\right){x}-805965a+4414455$ |
| 20.1-b1 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{12} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \) |
$3.718194980$ |
$1.772687765$ |
2.406765491 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -36\) , \( -140\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-36{x}-140$ |
| 20.1-b2 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \cdot 3 \) |
$1.239398326$ |
$15.95418988$ |
2.406765491 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 4\) , \( 4\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+4{x}+4$ |
| 20.1-b3 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.478796653$ |
$31.90837977$ |
2.406765491 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-{x}$ |
| 20.1-b4 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$7.436389961$ |
$3.545375530$ |
2.406765491 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -41\) , \( -116\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-41{x}-116$ |
| 20.1-c1 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{12} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3 \) |
$1.957359950$ |
$1.772687765$ |
3.800962354 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1598 a - 8746\) , \( -120126 a - 657954\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1598a-8746\right){x}-120126a-657954$ |
| 20.1-c2 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$0.652453316$ |
$15.95418988$ |
3.800962354 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 162 a + 894\) , \( 2298 a + 12590\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(162a+894\right){x}+2298a+12590$ |
| 20.1-c3 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.304906633$ |
$31.90837977$ |
3.800962354 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -58 a - 311\) , \( 519 a + 2846\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-58a-311\right){x}+519a+2846$ |
| 20.1-c4 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$3.914719901$ |
$3.545375530$ |
3.800962354 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1818 a - 9951\) , \( -94857 a - 519550\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1818a-9951\right){x}-94857a-519550$ |
| 20.1-d1 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.269743827$ |
$10.34365470$ |
3.056441960 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 15\) , \( 13\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+15{x}+13$ |
| 20.1-d2 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.269743827$ |
$10.34365470$ |
3.056441960 |
\( \frac{21296}{25} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 25\) , \( 25\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+25{x}+25$ |
| 20.1-d3 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.269743827$ |
$20.68730941$ |
3.056441960 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -5\) , \( -5\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-5{x}-5$ |
| 20.1-d4 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{30}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$2.07008$ |
$(2,a), (a+5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.269743827$ |
$20.68730941$ |
3.056441960 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -165\) , \( 763\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-165{x}+763$ |
*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.
