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 |
1083.1-a1 |
1083.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{2} \cdot 19^{8} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.637562893$ |
$2.262154252$ |
3.233482749 |
\( \frac{67419143}{390963} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -43 a + 118\) , \( 659 a - 1842\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-43a+118\right){x}+659a-1842$ |
1083.1-a2 |
1083.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{2} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.637562893$ |
$36.19446804$ |
3.233482749 |
\( \frac{389017}{57} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 7 a - 22\) , \( -11 a + 28\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(7a-22\right){x}-11a+28$ |
1083.1-a3 |
1083.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{4} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.275125787$ |
$9.048617011$ |
3.233482749 |
\( \frac{30664297}{3249} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 32 a - 92\) , \( 158 a - 444\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(32a-92\right){x}+158a-444$ |
1083.1-a4 |
1083.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{8} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$6.550251574$ |
$2.262154252$ |
3.233482749 |
\( \frac{115714886617}{1539} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -509 a - 914\) , \( -9754 a - 17480\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-509a-914\right){x}-9754a-17480$ |
1083.1-b1 |
1083.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{10} \) |
$2.34912$ |
$(-a+2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$9$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.085998375$ |
3.377949122 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -4390\) , \( -113432\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-4390{x}-113432$ |
1083.1-b2 |
1083.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{20} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$9$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.149959381$ |
3.377949122 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 20\) , \( -32\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+20{x}-32$ |
1083.1-c1 |
1083.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$3.679988085$ |
3.212156944 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 12 a - 31\) , \( 43 a - 121\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(12a-31\right){x}+43a-121$ |
1083.1-d1 |
1083.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.037574592$ |
$30.86363048$ |
1.012258976 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -2\) , \( 2\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-2{x}+2$ |
1083.1-e1 |
1083.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{10} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$0.190603013$ |
$3.435258684$ |
2.857653421 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 21952 a - 61463\) , \( -2700411 a + 7538465\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(21952a-61463\right){x}-2700411a+7538465$ |
1083.1-e2 |
1083.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{20} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.953015065$ |
$3.435258684$ |
2.857653421 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -98 a + 277\) , \( -861 a + 2405\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-98a+277\right){x}-861a+2405$ |
1083.1-f1 |
1083.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{2} \cdot 19^{8} \) |
$2.34912$ |
$(-a+2), (19)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.460486004$ |
$4.711524088$ |
3.787548392 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+8{x}+29$ |
1083.1-f2 |
1083.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{2} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.460486004$ |
$18.84609635$ |
3.787548392 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2{x}-1$ |
1083.1-f3 |
1083.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{4} \cdot 19^{4} \) |
$2.34912$ |
$(-a+2), (19)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.460486004$ |
$18.84609635$ |
3.787548392 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-7{x}+5$ |
1083.1-f4 |
1083.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1083.1 |
\( 3 \cdot 19^{2} \) |
\( 3^{8} \cdot 19^{2} \) |
$2.34912$ |
$(-a+2), (19)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.460486004$ |
$18.84609635$ |
3.787548392 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-102{x}+385$ |
*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.