| 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 |
| 2214.14-a1 |
2214.14-a |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{14} \cdot 3^{10} \cdot 41^{2} \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.319957630$ |
$2.640298093$ |
2.818392654 |
\( \frac{532819}{20172} a + \frac{50900933}{30258} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -12 a + 1\) , \( -4 a + 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-12a+1\right){x}-4a+50$ |
| 2214.14-a2 |
2214.14-a |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2 \cdot 3^{14} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.319957630$ |
$2.640298093$ |
2.818392654 |
\( -\frac{60343}{2214} a + \frac{5886962}{3321} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 4 a - 3\) , \( 2 a - 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(4a-3\right){x}+2a-6$ |
| 2214.14-a3 |
2214.14-a |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{13} \cdot 3^{8} \cdot 41^{4} \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.279830521$ |
$1.320149046$ |
2.818392654 |
\( -\frac{745561402649}{16954566} a + \frac{1625741146202}{8477283} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -117 a + 16\) , \( 584 a - 898\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-117a+16\right){x}+584a-898$ |
| 2214.14-a4 |
2214.14-a |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{16} \cdot 3^{8} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.319957630$ |
$2.640298093$ |
2.818392654 |
\( \frac{185510519}{1968} a + \frac{286349557}{984} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 21 a + 51\) , \( -66 a + 210\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(21a+51\right){x}-66a+210$ |
| 2214.14-b1 |
2214.14-b |
$2$ |
$2$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{17} \cdot 3^{20} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.394554901$ |
$1.141497073$ |
3.756456107 |
\( -\frac{8842795603}{8608032} a + \frac{23280637055}{4304016} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 78 a - 41\) , \( 178 a + 513\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(78a-41\right){x}+178a+513$ |
| 2214.14-b2 |
2214.14-b |
$2$ |
$2$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{10} \cdot 3^{13} \cdot 41^{2} \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.197277450$ |
$1.141497073$ |
3.756456107 |
\( \frac{346710555781}{139428864} a + \frac{582462068407}{69714432} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 23 a - 27\) , \( -50 a - 78\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(23a-27\right){x}-50a-78$ |
| 2214.14-c1 |
2214.14-c |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{14} \cdot 3^{13} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$0.634639890$ |
2.117305037 |
\( -\frac{7103601442567}{13284} a - \frac{1940239505125}{246} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 1103 a - 3892\) , \( -39400 a + 72566\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(1103a-3892\right){x}-39400a+72566$ |
| 2214.14-c2 |
2214.14-c |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{14} \cdot 3^{19} \cdot 41^{4} \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.634639890$ |
2.117305037 |
\( -\frac{68420885507089}{6006901006404} a + \frac{253981586057135}{3003450503202} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 73 a - 42\) , \( -932 a + 3546\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(73a-42\right){x}-932a+3546$ |
| 2214.14-c3 |
2214.14-c |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{20} \cdot 3^{10} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$2.538559561$ |
2.117305037 |
\( \frac{59063839}{283392} a + \frac{189438421}{141696} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 3 a - 32\) , \( -22 a - 40\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(3a-32\right){x}-22a-40$ |
| 2214.14-c4 |
2214.14-c |
$4$ |
$4$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{16} \cdot 3^{14} \cdot 41^{2} \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$1.269279780$ |
2.117305037 |
\( -\frac{740628481843}{19607184} a + \frac{261844623191}{9803592} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 68 a - 247\) , \( -700 a + 962\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(68a-247\right){x}-700a+962$ |
| 2214.14-d1 |
2214.14-d |
$1$ |
$1$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
2214.14 |
\( 2 \cdot 3^{3} \cdot 41 \) |
\( 2^{2} \cdot 3^{8} \cdot 41 \) |
$2.93966$ |
$(2,a+1), (3,a), (3,a+2), (41,a+25)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.844195512$ |
$3.954378786$ |
11.13723469 |
\( -\frac{4105879}{1476} a + \frac{5442179}{738} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -2 a + 6\) , \( 2 a - 1\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(-2a+6\right){x}+2a-1$ |
*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.
