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 |
27.2-a1 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.527418005$ |
$14.16516964$ |
2.087179461 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 31 a - 340\) , \( -1400 a + 10962\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(31w-340\right){x}-1400w+10962$ |
27.2-a2 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1.582254015$ |
$4.721723214$ |
2.087179461 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -39 a - 240\) , \( 430 a + 2846\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(-39w-240\right){x}+430w+2846$ |
27.2-a3 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$7.911270077$ |
$0.944344642$ |
2.087179461 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 391 a - 2980\) , \( 10278 a - 78708\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(391w-2980\right){x}+10278w-78708$ |
27.2-a4 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$2.637090025$ |
$2.833033928$ |
2.087179461 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 348 a - 2664\) , \( 13874 a - 106258\bigr] \) |
${y}^2+{y}={x}^3+\left(348w-2664\right){x}+13874w-106258$ |
27.2-b1 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$14.16516964$ |
1.319117816 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -15139 a - 100790\) , \( 2746408 a + 18288101\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-15139w-100790\right){x}+2746408w+18288101$ |
27.2-b2 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.721723214$ |
1.319117816 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -19 a + 160\) , \( 356 a - 2737\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-19w+160\right){x}+356w-2737$ |
27.2-b3 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.944344642$ |
1.319117816 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -19 a - 110\) , \( -394 a - 2617\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-19w-110\right){x}-394w-2617$ |
27.2-b4 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$1$ |
$2.833033928$ |
1.319117816 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 168 a + 1116\) , \( 10948 a + 72899\bigr] \) |
${y}^2+{y}={x}^3+\left(168w+1116\right){x}+10948w+72899$ |
*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.