| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.66936$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$1$ |
$41.83476262$ |
2.239363501 |
\( -201523200 a + 1983152128 \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 1210 a - 11907\) , \( -70425 a + 693036\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(1210a-11907\right){x}-70425a+693036$ |
| 1.1-a2 |
1.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.66936$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$1$ |
$41.83476262$ |
2.239363501 |
\( 201523200 a + 1781628928 \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -1210 a - 10697\) , \( 70425 a + 622611\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-1210a-10697\right){x}+70425a+622611$ |
| 3.1-a1 |
3.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$2.19701$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.256095187$ |
$17.28563035$ |
1.421756346 |
\( -\frac{1082}{27} a - \frac{8347}{27} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 11 a + 176\) , \( 41 a + 497\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(11a+176\right){x}+41a+497$ |
| 3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{3} \) |
$2.19701$ |
$(-a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.256095187$ |
$17.28563035$ |
1.421756346 |
\( \frac{1082}{27} a - \frac{3143}{9} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -11 a + 187\) , \( -41 a + 538\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11a+187\right){x}-41a+538$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$2.49628$ |
$(-6a+59)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.721374751$ |
$44.86879461$ |
6.930309305 |
\( \frac{8192}{25} a + \frac{126976}{25} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 26\) , \( -2 a - 6\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+26{x}-2a-6$ |
| 5.2-a1 |
5.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$2.49628$ |
$(-6a-53)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.721374751$ |
$44.86879461$ |
6.930309305 |
\( -\frac{8192}{25} a + \frac{135168}{25} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 26\) , \( 2 a - 8\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+26{x}+2a-8$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$2.89142$ |
$(-a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$2.363278325$ |
$6.466995931$ |
3.272387592 |
\( \frac{1082}{27} a - \frac{3143}{9} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 5439 a + 48112\) , \( -4091835 a - 36174948\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(5439a+48112\right){x}-4091835a-36174948$ |
| 9.2-b1 |
9.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.89142$ |
$(-a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$0.420243726$ |
$34.41427152$ |
1.548307088 |
\( -201523200 a + 1983152128 \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -281 a - 2452\) , \( 6158 a + 54406\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-281a-2452\right){x}+6158a+54406$ |
| 9.2-b2 |
9.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.89142$ |
$(-a+10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$2.101218632$ |
$6.882854304$ |
1.548307088 |
\( 201523200 a + 1781628928 \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 3929569 a - 38669962\) , \( 11553527768 a - 113695619204\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3929569a-38669962\right){x}+11553527768a-113695619204$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$2.89142$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$2.363278325$ |
$6.466995931$ |
3.272387592 |
\( -\frac{1082}{27} a - \frac{8347}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -5439 a + 53551\) , \( 4091835 a - 40266783\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5439a+53551\right){x}+4091835a-40266783$ |
| 9.3-b1 |
9.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.89142$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$2.101218632$ |
$6.882854304$ |
1.548307088 |
\( -201523200 a + 1983152128 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -3929569 a - 34740393\) , \( -11553527768 a - 102142091436\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-3929569a-34740393\right){x}-11553527768a-102142091436$ |
| 9.3-b2 |
9.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.89142$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 5$ |
2Cn, 5B |
$1$ |
\( 1 \) |
$0.420243726$ |
$34.41427152$ |
1.548307088 |
\( 201523200 a + 1781628928 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 281 a - 2733\) , \( -6158 a + 60564\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(281a-2733\right){x}-6158a+60564$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{6} \cdot 3^{3} \) |
$3.10704$ |
$(a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$7.075441879$ |
0.378739720 |
\( \frac{454507}{54} a + \frac{12943307}{216} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 111229 a - 1094584\) , \( 65337101 a - 642967468\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(111229a-1094584\right){x}+65337101a-642967468$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{6} \cdot 3^{7} \) |
$3.10704$ |
$(a+9), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$6.262690316$ |
$0.910260250$ |
4.272104205 |
\( -\frac{3885633216587533}{17496} a - \frac{8587996845360131}{4374} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 37 a - 365\) , \( 277 a - 2881\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(37a-365\right){x}+277a-2881$ |
| 12.1-c1 |
12.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{11} \) |
$3.10704$ |
$(a+9), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$4.447932144$ |
$5.537622453$ |
2.636931075 |
\( -\frac{1218738547}{354294} a - \frac{9907441841}{354294} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -48458796 a - 428413031\) , \( -569359275080 a - 5033574879883\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-48458796a-428413031\right){x}-569359275080a-5033574879883$ |
| 12.2-a1 |
12.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{6} \cdot 3^{3} \) |
$3.10704$ |
$(-a+10), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$7.075441879$ |
0.378739720 |
\( -\frac{454507}{54} a + \frac{4920445}{72} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -111230 a - 983354\) , \( -65337102 a - 577630366\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-111230a-983354\right){x}-65337102a-577630366$ |
| 12.2-b1 |
12.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{6} \cdot 3^{7} \) |
$3.10704$ |
$(-a+10), (2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 7 \) |
$1$ |
$0.910260250$ |
4.272104205 |
\( \frac{3885633216587533}{17496} a - \frac{12745873532676019}{5832} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -37 a - 328\) , \( -277 a - 2604\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-37a-328\right){x}-277a-2604$ |
| 12.2-c1 |
12.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{349}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{11} \) |
$3.10704$ |
$(-a+10), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$4.447932144$ |
$5.537622453$ |
2.636931075 |
\( \frac{1218738547}{354294} a - \frac{1854363398}{59049} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 48458794 a - 476871826\) , \( 569359275079 a - 5602934154962\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48458794a-476871826\right){x}+569359275079a-5602934154962$ |
*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.
