| 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 |
| 198.1-a1 |
198.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{8} \cdot 11^{8} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$4.813898015$ |
2.902889726 |
\( -\frac{192100033}{2371842} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -11\) , \( 69\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-11{x}+69$ |
| 198.1-a2 |
198.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.25559206$ |
2.902889726 |
\( \frac{912673}{528} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -1\) , \( -1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-{x}-1$ |
| 198.1-a3 |
198.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$19.25559206$ |
2.902889726 |
\( \frac{1180932193}{4356} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -21\) , \( 27\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-21{x}+27$ |
| 198.1-a4 |
198.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.25559206$ |
2.902889726 |
\( \frac{4824238966273}{66} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -351\) , \( 2337\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-351{x}+2337$ |
| 198.1-b1 |
198.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{6} \cdot 3^{4} \cdot 11^{12} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$7.724507840$ |
$0.635354791$ |
2.959516603 |
\( -\frac{7357983625}{127552392} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -41\) , \( -556\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-41{x}-556$ |
| 198.1-b2 |
198.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{12} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$2.574835946$ |
$5.718193122$ |
2.959516603 |
\( \frac{9938375}{176418} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}+20$ |
| 198.1-b3 |
198.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{6} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.287417973$ |
$22.87277248$ |
2.959516603 |
\( \frac{18609625}{1188} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -6\) , \( 4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-6{x}+4$ |
| 198.1-b4 |
198.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{12} \cdot 3^{2} \cdot 11^{6} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$3.862253920$ |
$2.541419165$ |
2.959516603 |
\( \frac{57736239625}{255552} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -81\) , \( -284\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-81{x}-284$ |
| 198.1-c1 |
198.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{4} \cdot 11^{20} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$3.028747336$ |
$0.884125736$ |
1.614770218 |
\( -\frac{112427521449300721}{466873642818} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -10058\) , \( 380252\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-10058{x}+380252$ |
| 198.1-c2 |
198.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{10} \cdot 3^{20} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.605749467$ |
$0.884125736$ |
1.614770218 |
\( \frac{168105213359}{228637728} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 112\) , \( -448\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+112{x}-448$ |
| 198.1-c3 |
198.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{20} \cdot 3^{10} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.302874733$ |
$3.536502944$ |
1.614770218 |
\( \frac{10091699281}{2737152} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -48\) , \( -128\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-48{x}-128$ |
| 198.1-c4 |
198.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{2} \cdot 11^{10} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$1.514373668$ |
$3.536502944$ |
1.614770218 |
\( \frac{112763292123580561}{1932612} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -10068\) , \( 379432\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-10068{x}+379432$ |
| 198.1-d1 |
198.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{4} \cdot 11^{20} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2^{4} \cdot 5 \) |
$7.225514002$ |
$0.056797834$ |
2.474766223 |
\( -\frac{112427521449300721}{466873642818} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -10055\) , \( -390309\bigr] \) |
${y}^2+{x}{y}={x}^{3}-10055{x}-390309$ |
| 198.1-d2 |
198.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{10} \cdot 3^{20} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{4} \cdot 5^{2} \) |
$1.445102800$ |
$1.419945868$ |
2.474766223 |
\( \frac{168105213359}{228637728} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 115\) , \( 561\bigr] \) |
${y}^2+{x}{y}={x}^{3}+115{x}+561$ |
| 198.1-d3 |
198.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{20} \cdot 3^{10} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{3} \cdot 5^{2} \) |
$0.722551400$ |
$5.679783475$ |
2.474766223 |
\( \frac{10091699281}{2737152} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -45\) , \( 81\bigr] \) |
${y}^2+{x}{y}={x}^{3}-45{x}+81$ |
| 198.1-d4 |
198.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{2} \cdot 11^{10} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2^{3} \cdot 5 \) |
$3.612757001$ |
$0.227191339$ |
2.474766223 |
\( \frac{112763292123580561}{1932612} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -10065\) , \( -389499\bigr] \) |
${y}^2+{x}{y}={x}^{3}-10065{x}-389499$ |
| 198.1-e1 |
198.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{6} \cdot 3^{4} \cdot 11^{12} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$0.183632440$ |
$2.444595345$ |
4.872620024 |
\( -\frac{7357983625}{127552392} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -38\) , \( 516\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-38{x}+516$ |
| 198.1-e2 |
198.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{12} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.550897322$ |
$2.444595345$ |
4.872620024 |
\( \frac{9938375}{176418} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 7\) , \( -15\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+7{x}-15$ |
| 198.1-e3 |
198.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{6} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.275448661$ |
$9.778381380$ |
4.872620024 |
\( \frac{18609625}{1188} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -3\) , \( -9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-3{x}-9$ |
| 198.1-e4 |
198.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{12} \cdot 3^{2} \cdot 11^{6} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.091816220$ |
$9.778381380$ |
4.872620024 |
\( \frac{57736239625}{255552} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -78\) , \( 204\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-78{x}+204$ |
| 198.1-f1 |
198.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{8} \cdot 11^{8} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.214828373$ |
2.930276290 |
\( -\frac{192100033}{2371842} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -12\) , \( -81\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-12{x}-81$ |
| 198.1-f2 |
198.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$19.43725397$ |
2.930276290 |
\( \frac{912673}{528} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2{x}-1$ |
| 198.1-f3 |
198.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{4} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$4.859313493$ |
2.930276290 |
\( \frac{1180932193}{4356} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -22\) , \( -49\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-22{x}-49$ |
| 198.1-f4 |
198.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
198.1 |
\( 2 \cdot 3^{2} \cdot 11 \) |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
$2.22347$ |
$(a+3), (a), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$1.214828373$ |
2.930276290 |
\( \frac{4824238966273}{66} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -352\) , \( -2689\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-352{x}-2689$ |
*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.
