| 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 |
| 14.1-a1 |
14.1-a |
$1$ |
$1$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.056313965$ |
$441.2877219$ |
2.996336534 |
\( -\frac{3112729}{784} a^{3} + \frac{825441}{392} a^{2} + \frac{26502365}{784} a - \frac{32146199}{784} \) |
\( \bigl[2 a^{3} + a^{2} - 11 a + 3\) , \( -a^{2} - a + 4\) , \( 1\) , \( -3 a^{3} - 2 a^{2} + 19 a + 4\) , \( -4 a^{3} - 2 a^{2} + 26 a\bigr] \) |
${y}^2+\left(2a^{3}+a^{2}-11a+3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-3a^{3}-2a^{2}+19a+4\right){x}-4a^{3}-2a^{2}+26a$ |
| 14.1-b1 |
14.1-b |
$2$ |
$2$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{34} \cdot 7^{2} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 17 \) |
$0.054533573$ |
$179.5662673$ |
5.017995604 |
\( -\frac{1402574522628382639}{841813590016} a^{3} - \frac{348633256698689353}{420906795008} a^{2} + \frac{8799534999868774715}{841813590016} a - \frac{844138674551993185}{841813590016} \) |
\( \bigl[a\) , \( -a^{2} - a + 3\) , \( a^{3} + a^{2} - 4 a - 1\) , \( -4 a^{3} + 30 a - 32\) , \( 5 a^{3} - 7 a^{2} - 28 a + 41\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-4a^{3}+30a-32\right){x}+5a^{3}-7a^{2}-28a+41$ |
| 14.1-b2 |
14.1-b |
$2$ |
$2$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{17} \cdot 7^{4} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 17 \) |
$0.109067146$ |
$179.5662673$ |
5.017995604 |
\( -\frac{705345550725079}{314703872} a^{3} + \frac{1343251625907103}{157351936} a^{2} - \frac{2604379864571517}{314703872} a + \frac{252085497925447}{314703872} \) |
\( \bigl[a^{3} - 6 a + 4\) , \( -a^{3} - a^{2} + 6 a + 2\) , \( 2 a^{3} + a^{2} - 11 a + 2\) , \( 106 a^{3} + 127 a^{2} - 455 a + 56\) , \( 1513 a^{3} + 1834 a^{2} - 6526 a + 689\bigr] \) |
${y}^2+\left(a^{3}-6a+4\right){x}{y}+\left(2a^{3}+a^{2}-11a+2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a+2\right){x}^{2}+\left(106a^{3}+127a^{2}-455a+56\right){x}+1513a^{3}+1834a^{2}-6526a+689$ |
| 14.1-c1 |
14.1-c |
$1$ |
$1$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.018230905$ |
$1191.147189$ |
5.236683102 |
\( -\frac{3112729}{784} a^{3} + \frac{825441}{392} a^{2} + \frac{26502365}{784} a - \frac{32146199}{784} \) |
\( \bigl[2 a^{3} + a^{2} - 10 a + 2\) , \( a^{3} + a^{2} - 6 a - 1\) , \( 2 a^{3} + a^{2} - 11 a + 3\) , \( 8 a^{3} + 9 a^{2} - 40 a + 4\) , \( 19 a^{3} + 23 a^{2} - 85 a + 8\bigr] \) |
${y}^2+\left(2a^{3}+a^{2}-10a+2\right){x}{y}+\left(2a^{3}+a^{2}-11a+3\right){y}={x}^{3}+\left(a^{3}+a^{2}-6a-1\right){x}^{2}+\left(8a^{3}+9a^{2}-40a+4\right){x}+19a^{3}+23a^{2}-85a+8$ |
| 14.1-d1 |
14.1-d |
$2$ |
$2$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{17} \cdot 7^{4} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.810611253$ |
$76.91955795$ |
1.879498800 |
\( -\frac{705345550725079}{314703872} a^{3} + \frac{1343251625907103}{157351936} a^{2} - \frac{2604379864571517}{314703872} a + \frac{252085497925447}{314703872} \) |
\( \bigl[1\) , \( -2 a^{3} - a^{2} + 12 a - 2\) , \( 2 a^{3} + a^{2} - 11 a + 2\) , \( -7 a^{3} + 47 a - 29\) , \( 2 a^{3} - 3 a^{2} - 13 a + 17\bigr] \) |
${y}^2+{x}{y}+\left(2a^{3}+a^{2}-11a+2\right){y}={x}^{3}+\left(-2a^{3}-a^{2}+12a-2\right){x}^{2}+\left(-7a^{3}+47a-29\right){x}+2a^{3}-3a^{2}-13a+17$ |
| 14.1-d2 |
14.1-d |
$2$ |
$2$ |
4.4.17609.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{34} \cdot 7^{2} \) |
$16.49195$ |
$(a^3+a^2-5a+1), (-a^3+6a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.621222506$ |
$38.45977897$ |
1.879498800 |
\( -\frac{1402574522628382639}{841813590016} a^{3} - \frac{348633256698689353}{420906795008} a^{2} + \frac{8799534999868774715}{841813590016} a - \frac{844138674551993185}{841813590016} \) |
\( \bigl[2 a^{3} + a^{2} - 11 a + 3\) , \( a^{3} + a^{2} - 5 a - 2\) , \( a^{3} - 6 a + 3\) , \( 986 a^{3} + 480 a^{2} - 6186 a + 662\) , \( 25024 a^{3} + 12147 a^{2} - 157124 a + 16845\bigr] \) |
${y}^2+\left(2a^{3}+a^{2}-11a+3\right){x}{y}+\left(a^{3}-6a+3\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-2\right){x}^{2}+\left(986a^{3}+480a^{2}-6186a+662\right){x}+25024a^{3}+12147a^{2}-157124a+16845$ |
*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.
