| 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 |
| 196.1-a1 |
196.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.060084072$ |
$31.58652452$ |
2.963939123 |
\( -\frac{1492553}{224} a + \frac{862213}{32} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( a - 6\) , \( -2 a + 4\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-6\right){x}-2a+4$ |
| 196.1-b1 |
196.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$24.09859934$ |
3.763568915 |
\( -\frac{32481205}{3584} a + \frac{18492629}{512} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -18 a - 44\) , \( -27 a - 71\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-18a-44\right){x}-27a-71$ |
| 196.1-b2 |
196.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.677622149$ |
3.763568915 |
\( \frac{265362304805}{343} a + \frac{5735274222913}{2744} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -1243 a - 3354\) , \( -41647 a - 112511\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1243a-3354\right){x}-41647a-112511$ |
| 196.1-c1 |
196.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{28} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 23 \) |
$0.085598703$ |
$6.241254947$ |
3.838000373 |
\( -\frac{275903901235}{58720256} a + \frac{90394778161}{4194304} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 7613 a - 28178\) , \( 640517 a - 2370914\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7613a-28178\right){x}+640517a-2370914$ |
| 196.1-d1 |
196.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.416440697$ |
$13.08135831$ |
1.929159121 |
\( -\frac{5629733915}{702464} a + \frac{11194820639}{351232} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -212 a - 570\) , \( 1122 a + 3030\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-212a-570\right){x}+1122a+3030$ |
| 196.1-d2 |
196.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{36} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$4.249322093$ |
$1.453484256$ |
1.929159121 |
\( \frac{139808166166932965}{60129542144} a + \frac{190488170863162399}{30064771072} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -8542 a - 23075\) , \( -780155 a - 2107638\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-8542a-23075\right){x}-780155a-2107638$ |
| 196.1-e1 |
196.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.416440697$ |
$13.08135831$ |
1.929159121 |
\( \frac{5629733915}{702464} a + \frac{16759907363}{702464} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 211 a - 781\) , \( -1123 a + 4153\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(211a-781\right){x}-1123a+4153$ |
| 196.1-e2 |
196.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{36} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$4.249322093$ |
$1.453484256$ |
1.929159121 |
\( -\frac{139808166166932965}{60129542144} a + \frac{520784507893257763}{60129542144} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 8541 a - 31616\) , \( 780154 a - 2887792\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8541a-31616\right){x}+780154a-2887792$ |
| 196.1-f1 |
196.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{50} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5^{4} \) |
$1$ |
$1.603614252$ |
6.261061760 |
\( \frac{12572781605371889}{11509170176} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 3100387 a - 11476272\) , \( -5310791464 a + 19658224497\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3100387a-11476272\right){x}-5310791464a+19658224497$ |
| 196.1-f2 |
196.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{30} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 5^{2} \) |
$1$ |
$0.064144570$ |
6.261061760 |
\( \frac{222726737088758142449}{151921968318176} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 80821987 a - 299167632\) , \( 707317798136 a - 2618180767503\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(80821987a-299167632\right){x}+707317798136a-2618180767503$ |
| 196.1-g1 |
196.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{28} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 23 \) |
$0.085598703$ |
$6.241254947$ |
3.838000373 |
\( \frac{275903901235}{58720256} a + \frac{989622993019}{58720256} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -7613 a - 20565\) , \( -640517 a - 1730397\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-7613a-20565\right){x}-640517a-1730397$ |
| 196.1-h1 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{36} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$12.83949377$ |
$0.436190660$ |
1.749292084 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 196.1-h2 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.426610419$ |
$35.33144352$ |
1.749292084 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 196.1-h3 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.279831257$ |
$3.925715946$ |
1.749292084 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 196.1-h4 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{12} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$8.559662515$ |
$3.925715946$ |
1.749292084 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 196.1-h5 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 7^{4} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.853220838$ |
$35.33144352$ |
1.749292084 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 196.1-h6 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{18} \cdot 7^{4} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$25.67898754$ |
$0.436190660$ |
1.749292084 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
| 196.1-i1 |
196.1-i |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.677622149$ |
3.763568915 |
\( -\frac{265362304805}{343} a + \frac{1122596094479}{392} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 1243 a - 4597\) , \( 41647 a - 154158\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(1243a-4597\right){x}+41647a-154158$ |
| 196.1-i2 |
196.1-i |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$24.09859934$ |
3.763568915 |
\( \frac{32481205}{3584} a + \frac{48483599}{1792} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 18 a - 62\) , \( 27 a - 98\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(18a-62\right){x}+27a-98$ |
| 196.1-j1 |
196.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$2.14089$ |
$(-a+4), (a+3), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.060084072$ |
$31.58652452$ |
2.963939123 |
\( \frac{1492553}{224} a + \frac{2271469}{112} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -2 a - 5\) , \( a + 2\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a-5\right){x}+a+2$ |
*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.
