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 |
1444.1-a1 |
1444.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{10} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$1$ |
$5.550558930$ |
6.056156297 |
\( -\frac{37966934881}{4952198} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 352 a - 976\) , \( -5990 a + 16723\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(352a-976\right){x}-5990a+16723$ |
1444.1-a2 |
1444.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$5.550558930$ |
6.056156297 |
\( -\frac{1}{608} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 2 a + 4\) , \( 30 a - 77\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+4\right){x}+30a-77$ |
1444.1-b1 |
1444.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{4} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.560781336$ |
$26.71213644$ |
3.268831455 |
\( \frac{132651}{76} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 1\) , \( -a - 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+{x}-a-2$ |
1444.1-b2 |
1444.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{4} \) |
$2.52429$ |
$(2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.121562673$ |
$13.35606822$ |
3.268831455 |
\( \frac{149721291}{722} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -10 a - 19\) , \( 15 a + 26\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-10a-19\right){x}+15a+26$ |
1444.1-c1 |
1444.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{6} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$32.17041206$ |
2.340053149 |
\( -\frac{413493625}{152} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -16\) , \( 22\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-16{x}+22$ |
1444.1-c2 |
1444.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{54} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{3} \) |
$1$ |
$0.397165580$ |
2.340053149 |
\( -\frac{69173457625}{2550136832} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -86\) , \( -2456\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-86{x}-2456$ |
1444.1-c3 |
1444.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{18} \cdot 19^{6} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$3.574490228$ |
2.340053149 |
\( \frac{94196375}{3511808} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9\) , \( 90\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+9{x}+90$ |
1444.1-d1 |
1444.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{6} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$1.446313524$ |
0.946834457 |
\( -\frac{413493625}{152} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -79 a - 140\) , \( -612 a - 1097\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-79a-140\right){x}-612a-1097$ |
1444.1-d2 |
1444.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{54} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$1.446313524$ |
0.946834457 |
\( -\frac{69173457625}{2550136832} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 427 a - 1198\) , \( -58511 a + 163337\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(427a-1198\right){x}-58511a+163337$ |
1444.1-d3 |
1444.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{18} \cdot 19^{6} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$1.446313524$ |
0.946834457 |
\( \frac{94196375}{3511808} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -48 a + 132\) , \( 2118 a - 5915\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-48a+132\right){x}+2118a-5915$ |
1444.1-e1 |
1444.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{4} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.560781336$ |
$26.71213644$ |
3.268831455 |
\( \frac{132651}{76} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 5 a + 4\) , \( 3 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+4\right){x}+3a+6$ |
1444.1-e2 |
1444.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{4} \) |
$2.52429$ |
$(2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.121562673$ |
$13.35606822$ |
3.268831455 |
\( \frac{149721291}{722} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 15 a - 26\) , \( -33 a + 100\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15a-26\right){x}-33a+100$ |
1444.1-f1 |
1444.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{10} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 5 \) |
$1$ |
$0.671163407$ |
0.732299313 |
\( -\frac{37966934881}{4952198} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -70\) , \( -279\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-70{x}-279$ |
1444.1-f2 |
1444.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1444.1 |
\( 2^{2} \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{2} \) |
$2.52429$ |
$(2), (19)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$16.77908518$ |
0.732299313 |
\( -\frac{1}{608} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+1$ |
*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.