| 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 |
| 3169.1-a1 |
3169.1-a |
$4$ |
$6$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( - 3169^{6} \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$2.028450013$ |
$5.705335007$ |
2.578859653 |
\( \frac{26506877180386585664122900665}{1012822720258273404481} a^{3} - \frac{39073726452891123999646595975}{1012822720258273404481} a^{2} - \frac{60793069622185379945676733510}{1012822720258273404481} a + \frac{55460499812512462719654534663}{1012822720258273404481} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - 2 a + 1\) , \( -5 a^{3} + 3 a^{2} + 36 a - 85\) , \( -136 a^{3} + 320 a^{2} + 82 a - 424\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-5a^{3}+3a^{2}+36a-85\right){x}-136a^{3}+320a^{2}+82a-424$ |
| 3169.1-a2 |
3169.1-a |
$4$ |
$6$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( 3169^{3} \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1.014225006$ |
$22.82134003$ |
2.578859653 |
\( \frac{352591025595071893365}{31824875809} a^{3} + \frac{386542461602387052175}{31824875809} a^{2} - \frac{247945033312905181810}{31824875809} a - \frac{168049691720730710378}{31824875809} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - 2 a + 1\) , \( 15 a^{3} - 12 a^{2} - 49 a - 25\) , \( 62 a^{3} - 38 a^{2} - 199 a - 85\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(15a^{3}-12a^{2}-49a-25\right){x}+62a^{3}-38a^{2}-199a-85$ |
| 3169.1-a3 |
3169.1-a |
$4$ |
$6$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( 3169 \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$0.338075002$ |
$1848.528542$ |
2.578859653 |
\( -\frac{919454893}{3169} a^{3} + \frac{2661427405}{3169} a^{2} - \frac{1572449138}{3169} a + \frac{87031533}{3169} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - 2 a + 1\) , \( -2 a^{2} + a\) , \( -a^{3}\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-2a^{2}+a\right){x}-a^{3}$ |
| 3169.1-a4 |
3169.1-a |
$4$ |
$6$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( - 3169^{2} \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.676150004$ |
$462.1321356$ |
2.578859653 |
\( -\frac{13920283943200893235}{10042561} a^{3} + \frac{32791655078311352917}{10042561} a^{2} - \frac{2693951987993741684}{10042561} a - \frac{10268162489949316762}{10042561} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - 2 a + 1\) , \( 15 a^{3} - 42 a^{2} + 11 a + 15\) , \( -66 a^{3} + 150 a^{2} - 7 a - 45\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(15a^{3}-42a^{2}+11a+15\right){x}-66a^{3}+150a^{2}-7a-45$ |
| 3169.1-b1 |
3169.1-b |
$2$ |
$2$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( - 3169^{2} \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.077142659$ |
$490.4667147$ |
2.810380608 |
\( -\frac{31169334220237}{10042561} a^{3} + \frac{8914525238256}{10042561} a^{2} + \frac{94831115348295}{10042561} a + \frac{48887083672202}{10042561} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 1\) , \( -a^{3} + 3 a\) , \( a\) , \( 5 a^{3} - 11 a^{2} + a + 1\) , \( 10 a^{3} - 23 a^{2} + 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(5a^{3}-11a^{2}+a+1\right){x}+10a^{3}-23a^{2}+9$ |
| 3169.1-b2 |
3169.1-b |
$2$ |
$2$ |
4.4.725.1 |
$4$ |
$[4, 0]$ |
3169.1 |
\( 3169 \) |
\( 3169 \) |
$6.59057$ |
$(-a^3-3a^2+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.038571329$ |
$1961.866858$ |
2.810380608 |
\( \frac{2653932}{3169} a^{3} + \frac{5992969}{3169} a^{2} + \frac{3082469}{3169} a + \frac{6495478}{3169} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 1\) , \( -a^{3} + 3 a\) , \( a\) , \( -a^{2} + a + 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-a^{2}+a+1\right){x}$ |
*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.
