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 |
18.1-a1 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$5.365735541$ |
2.154222274 |
\( -\frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -6112 a - 26627\) , \( -275601 a - 1201300\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6112a-26627\right){x}-275601a-1201300$ |
18.1-a2 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$2.682867770$ |
2.154222274 |
\( \frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -46792 a - 203947\) , \( 11319975 a + 49342644\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-46792a-203947\right){x}+11319975a+49342644$ |
18.1-b1 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$2.682867770$ |
2.154222274 |
\( -\frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 46791 a - 203947\) , \( -11319976 a + 49342644\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(46791a-203947\right){x}-11319976a+49342644$ |
18.1-b2 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$5.365735541$ |
2.154222274 |
\( \frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 6111 a - 26627\) , \( 275600 a - 1201300\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6111a-26627\right){x}+275600a-1201300$ |
18.1-c1 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 11 \) |
$0.134555585$ |
$10.28817665$ |
1.746731012 |
\( -\frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -6110 a - 26622\) , \( 269490 a + 1174670\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6110a-26622\right){x}+269490a+1174670$ |
18.1-c2 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 11 \) |
$0.269111170$ |
$2.572044164$ |
1.746731012 |
\( \frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -46790 a - 203942\) , \( -11366766 a - 49546594\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-46790a-203942\right){x}-11366766a-49546594$ |
18.1-d1 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 11 \) |
$0.269111170$ |
$2.572044164$ |
1.746731012 |
\( -\frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 46789 a - 203942\) , \( 11366766 a - 49546594\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(46789a-203942\right){x}+11366766a-49546594$ |
18.1-d2 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 11 \) |
$0.134555585$ |
$10.28817665$ |
1.746731012 |
\( \frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 6109 a - 26622\) , \( -269490 a + 1174670\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6109a-26622\right){x}-269490a+1174670$ |
*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.