| 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 |
| 153.2-a1 |
153.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{3} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$5.722265628$ |
2.497401465 |
\( -\frac{38546445}{4913} a - \frac{64561539}{4913} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2 a - 4\) , \( -2 a - 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2a-4\right){x}-2a-4$ |
| 153.2-a2 |
153.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{9} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$5.722265628$ |
2.497401465 |
\( \frac{10935}{17} a - \frac{5616}{17} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 7 a - 17\) , \( 21 a - 57\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(7a-17\right){x}+21a-57$ |
| 153.2-b1 |
153.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( - 3^{6} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$9.864496947$ |
2.152609712 |
\( -\frac{20811}{17} a + \frac{1220}{17} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -2 a\) , \( -a - 2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}-2a{x}-a-2$ |
| 153.2-b2 |
153.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( - 3^{6} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$3.288165649$ |
2.152609712 |
\( -\frac{2723256379739}{4913} a + \frac{7601399159655}{4913} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 13 a\) , \( 23 a - 41\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+13a{x}+23a-41$ |
| 153.2-c1 |
153.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{9} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.628156996$ |
$11.02633562$ |
2.015247996 |
\( -\frac{38546445}{4913} a - \frac{64561539}{4913} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -5 a + 9\) , \( -4 a + 13\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-5a+9\right){x}-4a+13$ |
| 153.2-c2 |
153.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{3} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.209385665$ |
$11.02633562$ |
2.015247996 |
\( \frac{10935}{17} a - \frac{5616}{17} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -1\) , \( -2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-{x}-2$ |
| 153.2-d1 |
153.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{9} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.687988911$ |
1.173134538 |
\( -\frac{38546445}{4913} a - \frac{64561539}{4913} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -19 a - 34\) , \( -91 a - 164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-19a-34\right){x}-91a-164$ |
| 153.2-d2 |
153.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{3} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$24.19190020$ |
1.173134538 |
\( \frac{10935}{17} a - \frac{5616}{17} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( a + 1\) , \( a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}+a+1$ |
| 153.2-e1 |
153.2-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( - 3^{6} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.162669075$ |
$29.95731825$ |
2.126807977 |
\( -\frac{20811}{17} a + \frac{1220}{17} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( a - 8\) , \( -2 a + 2\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a-8\right){x}-2a+2$ |
| 153.2-e2 |
153.2-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( - 3^{6} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.488007227$ |
$9.985772750$ |
2.126807977 |
\( -\frac{2723256379739}{4913} a + \frac{7601399159655}{4913} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 136 a - 383\) , \( -1325 a + 3695\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(136a-383\right){x}-1325a+3695$ |
| 153.2-f1 |
153.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{3} \cdot 17^{3} \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.045678007$ |
$15.53860121$ |
1.858620291 |
\( -\frac{38546445}{4913} a - \frac{64561539}{4913} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -7 a + 14\) , \( 9 a - 27\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a+14\right){x}+9a-27$ |
| 153.2-f2 |
153.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
153.2 |
\( 3^{2} \cdot 17 \) |
\( 3^{9} \cdot 17 \) |
$1.44019$ |
$(-a+2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.137034023$ |
$15.53860121$ |
1.858620291 |
\( \frac{10935}{17} a - \frac{5616}{17} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+3{x}+1$ |
*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.
