| 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 |
| 3.1-a1 |
3.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{5} \) |
$1.30957$ |
$(2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$20.10785757$ |
1.805738916 |
\( -\frac{53851}{243} a + \frac{299782}{243} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 121029 a - 673856\) , \( -29861671 a + 166262744\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(121029a-673856\right){x}-29861671a+166262744$ |
| 3.1-b1 |
3.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{5} \) |
$1.30957$ |
$(2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.086621101$ |
$30.24536113$ |
2.352727526 |
\( -\frac{53851}{243} a + \frac{299782}{243} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 121025 a - 673841\) , \( 30103726 a - 167610453\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(121025a-673841\right){x}+30103726a-167610453$ |
| 3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{5} \) |
$1.30957$ |
$(2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$20.10785757$ |
1.805738916 |
\( \frac{53851}{243} a + \frac{299782}{243} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -121030 a - 673856\) , \( 29861671 a + 166262744\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-121030a-673856\right){x}+29861671a+166262744$ |
| 3.2-b1 |
3.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{5} \) |
$1.30957$ |
$(2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.086621101$ |
$30.24536113$ |
2.352727526 |
\( \frac{53851}{243} a + \frac{299782}{243} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -121026 a - 673841\) , \( -30103726 a - 167610453\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-121026a-673841\right){x}-30103726a-167610453$ |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.40723$ |
$(7a+39)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.207269743$ |
$17.46386550$ |
1.950368601 |
\( -256 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -276640 a - 1540256\) , \( -519754262 a - 2893869251\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-276640a-1540256\right){x}-519754262a-2893869251$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.40723$ |
$(7a+39)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.207269743$ |
$17.46386550$ |
1.950368601 |
\( -256 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 276640 a - 1540256\) , \( 519754262 a - 2893869251\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(276640a-1540256\right){x}+519754262a-2893869251$ |
| 5.1-a1 |
5.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$1.48796$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.683574669$ |
$4.528902905$ |
1.369444845 |
\( \frac{130527768}{625} a - \frac{726739513}{625} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 5113 a + 28478\) , \( -5333193 a - 29693963\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5113a+28478\right){x}-5333193a-29693963$ |
| 5.1-a2 |
5.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$1.48796$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.367149338$ |
$4.528902905$ |
1.369444845 |
\( -\frac{12149809152051}{25} a + \frac{67647275026916}{25} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -177542 a - 988502\) , \( -92274555 a - 513762980\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-177542a-988502\right){x}-92274555a-513762980$ |
| 5.1-b1 |
5.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$1.48796$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.238653098$ |
$33.90186147$ |
1.453147754 |
\( \frac{130527768}{625} a - \frac{726739513}{625} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 5117 a + 28478\) , \( 5343423 a + 29750915\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5117a+28478\right){x}+5343423a+29750915$ |
| 5.1-b2 |
5.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$1.48796$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.477306196$ |
$33.90186147$ |
1.453147754 |
\( -\frac{12149809152051}{25} a + \frac{67647275026916}{25} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -177538 a - 988502\) , \( 91919475 a + 511785972\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-177538a-988502\right){x}+91919475a+511785972$ |
| 5.2-a1 |
5.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{4} \) |
$1.48796$ |
$(a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.683574669$ |
$4.528902905$ |
1.369444845 |
\( -\frac{130527768}{625} a - \frac{726739513}{625} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -5114 a + 28478\) , \( 5333192 a - 29693963\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5114a+28478\right){x}+5333192a-29693963$ |
| 5.2-a2 |
5.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$1.48796$ |
$(a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.367149338$ |
$4.528902905$ |
1.369444845 |
\( \frac{12149809152051}{25} a + \frac{67647275026916}{25} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 177541 a - 988502\) , \( 92274554 a - 513762980\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(177541a-988502\right){x}+92274554a-513762980$ |
| 5.2-b1 |
5.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{4} \) |
$1.48796$ |
$(a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.238653098$ |
$33.90186147$ |
1.453147754 |
\( -\frac{130527768}{625} a - \frac{726739513}{625} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -5118 a + 28478\) , \( -5343424 a + 29750915\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5118a+28478\right){x}-5343424a+29750915$ |
| 5.2-b2 |
5.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$1.48796$ |
$(a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.477306196$ |
$33.90186147$ |
1.453147754 |
\( \frac{12149809152051}{25} a + \frac{67647275026916}{25} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 177537 a - 988502\) , \( -91919476 a + 511785972\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(177537a-988502\right){x}-91919476a+511785972$ |
| 6.1-a1 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{4} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$23.14294041$ |
4.156594802 |
\( -\frac{53375}{162} a - \frac{120632}{81} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -3 a + 30\) , \( 10 a - 65\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a+30\right){x}+10a-65$ |
| 6.1-a2 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{8} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$23.14294041$ |
4.156594802 |
\( \frac{19037336423}{13122} a + \frac{106078120759}{13122} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 32 a - 165\) , \( 243 a - 1362\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(32a-165\right){x}+243a-1362$ |
| 6.1-b1 |
6.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$6.424132785$ |
1.153808309 |
\( -\frac{460264183}{108} a + \frac{1281314225}{54} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 442 a - 2456\) , \( 11044 a - 61494\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(442a-2456\right){x}+11044a-61494$ |
| 6.1-c1 |
6.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.047228043$ |
$36.82733723$ |
1.874306778 |
\( -\frac{460264183}{108} a + \frac{1281314225}{54} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 438 a - 2441\) , \( -10163 a + 56585\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(438a-2441\right){x}-10163a+56585$ |
| 6.1-d1 |
6.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{4} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.026186464$ |
$12.01442843$ |
2.214361641 |
\( -\frac{53375}{162} a - \frac{120632}{81} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( a + 45\) , \( -12 a + 136\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+45\right){x}-12a+136$ |
| 6.1-d2 |
6.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{8} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.052372928$ |
$12.01442843$ |
2.214361641 |
\( \frac{19037336423}{13122} a + \frac{106078120759}{13122} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 36 a - 150\) , \( -175 a + 1043\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(36a-150\right){x}-175a+1043$ |
| 6.1-e1 |
6.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3 \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.541589431$ |
$15.34539498$ |
2.985364758 |
\( -\frac{1390604605371112501}{3} a - \frac{15485117528557232989}{6} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -11433 a - 63662\) , \( 1524244 a + 8486635\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-11433a-63662\right){x}+1524244a+8486635$ |
| 6.1-e2 |
6.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{10} \cdot 3^{5} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$0.108317886$ |
$15.34539498$ |
2.985364758 |
\( -\frac{17864401}{3888} a + \frac{161458571}{7776} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -13 a - 72\) , \( 44 a + 245\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-13a-72\right){x}+44a+245$ |
| 6.1-f1 |
6.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3 \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$0.382055982$ |
1.715482001 |
\( -\frac{1390604605371112501}{3} a - \frac{15485117528557232989}{6} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -11437 a - 63677\) , \( -1547114 a - 8613978\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11437a-63677\right){x}-1547114a-8613978$ |
| 6.1-f2 |
6.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{10} \cdot 3^{5} \) |
$1.55735$ |
$(7a+39), (2a+11)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$9.551399552$ |
1.715482001 |
\( -\frac{17864401}{3888} a + \frac{161458571}{7776} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -17 a - 87\) , \( -74 a - 408\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-17a-87\right){x}-74a-408$ |
| 6.2-a1 |
6.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{4} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$23.14294041$ |
4.156594802 |
\( \frac{53375}{162} a - \frac{120632}{81} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 3 a + 30\) , \( -10 a - 65\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a+30\right){x}-10a-65$ |
| 6.2-a2 |
6.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{8} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$23.14294041$ |
4.156594802 |
\( -\frac{19037336423}{13122} a + \frac{106078120759}{13122} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -32 a - 165\) , \( -243 a - 1362\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-32a-165\right){x}-243a-1362$ |
| 6.2-b1 |
6.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$6.424132785$ |
1.153808309 |
\( \frac{460264183}{108} a + \frac{1281314225}{54} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -443 a - 2456\) , \( -11044 a - 61494\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-443a-2456\right){x}-11044a-61494$ |
| 6.2-c1 |
6.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.047228043$ |
$36.82733723$ |
1.874306778 |
\( \frac{460264183}{108} a + \frac{1281314225}{54} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -439 a - 2441\) , \( 10163 a + 56585\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-439a-2441\right){x}+10163a+56585$ |
| 6.2-d1 |
6.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3^{4} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.026186464$ |
$12.01442843$ |
2.214361641 |
\( \frac{53375}{162} a - \frac{120632}{81} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -a + 45\) , \( 12 a + 136\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a+45\right){x}+12a+136$ |
| 6.2-d2 |
6.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2 \cdot 3^{8} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.052372928$ |
$12.01442843$ |
2.214361641 |
\( -\frac{19037336423}{13122} a + \frac{106078120759}{13122} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -36 a - 150\) , \( 175 a + 1043\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-36a-150\right){x}+175a+1043$ |
| 6.2-e1 |
6.2-e |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{10} \cdot 3^{5} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$0.108317886$ |
$15.34539498$ |
2.985364758 |
\( \frac{17864401}{3888} a + \frac{161458571}{7776} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 13 a - 72\) , \( -44 a + 245\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(13a-72\right){x}-44a+245$ |
| 6.2-e2 |
6.2-e |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3 \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.541589431$ |
$15.34539498$ |
2.985364758 |
\( \frac{1390604605371112501}{3} a - \frac{15485117528557232989}{6} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 11433 a - 63662\) , \( -1524244 a + 8486635\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11433a-63662\right){x}-1524244a+8486635$ |
| 6.2-f1 |
6.2-f |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{10} \cdot 3^{5} \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$9.551399552$ |
1.715482001 |
\( \frac{17864401}{3888} a + \frac{161458571}{7776} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 17 a - 87\) , \( 74 a - 408\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(17a-87\right){x}+74a-408$ |
| 6.2-f2 |
6.2-f |
$2$ |
$5$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3 \) |
$1.55735$ |
$(7a+39), (2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$0.382055982$ |
1.715482001 |
\( \frac{1390604605371112501}{3} a - \frac{15485117528557232989}{6} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 11437 a - 63677\) , \( 1547114 a - 8613978\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(11437a-63677\right){x}+1547114a-8613978$ |
| 8.1-a1 |
8.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$6.969635407$ |
2.503566944 |
\( -76995328 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -18534880 a - 103197834\) , \( 102527132824 a + 570846916367\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-18534880a-103197834\right){x}+102527132824a+570846916367$ |
| 8.1-b1 |
8.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$6.969635407$ |
2.503566944 |
\( -76995328 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -18534880 a - 103197834\) , \( -102527132824 a - 570846916367\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-18534880a-103197834\right){x}-102527132824a-570846916367$ |
| 8.1-c1 |
8.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.97825510$ |
0.762346159 |
\( -1372 \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2 a + 13\) , \( 4 a + 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2a+13\right){x}+4a+25$ |
| 8.1-c2 |
8.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.97825510$ |
0.762346159 |
\( 4096766 \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2 a + 3\) , \( -5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2a+3\right){x}-5$ |
| 8.1-d1 |
8.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.97825510$ |
0.762346159 |
\( -1372 \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -4 a + 13\) , \( -5 a + 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+13\right){x}-5a+25$ |
| 8.1-d2 |
8.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.67349$ |
$(7a+39)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.97825510$ |
0.762346159 |
\( 4096766 \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -4 a + 3\) , \( -a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+3\right){x}-a-5$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$4$ |
\( 2 \) |
$1$ |
$6.753600901$ |
4.851930119 |
\( 820200087 a + 4566677886 \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 882579 a - 4913995\) , \( -1343089295 a + 7478004702\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(882579a-4913995\right){x}-1343089295a+7478004702$ |
| 9.2-b1 |
9.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a-11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.451472199$ |
1.697534518 |
\( 820200087 a + 4566677886 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 882579 a - 4913995\) , \( 1344854453 a - 7487832696\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(882579a-4913995\right){x}+1344854453a-7487832696$ |
| 9.2-c1 |
9.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.06155021$ |
0.766086219 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -508 a - 2841\) , \( -2970 a - 16544\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-508a-2841\right){x}-2970a-16544$ |
| 9.2-c2 |
9.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.06155021$ |
0.766086219 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 252 a - 1321\) , \( 70 a - 204\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(252a-1321\right){x}+70a-204$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$4$ |
\( 2 \) |
$1$ |
$6.753600901$ |
4.851930119 |
\( -820200087 a + 4566677886 \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -882580 a - 4913995\) , \( 1343089294 a + 7478004702\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-882580a-4913995\right){x}+1343089294a+7478004702$ |
| 9.3-b1 |
9.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.451472199$ |
1.697534518 |
\( -820200087 a + 4566677886 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -882580 a - 4913995\) , \( -1344854454 a - 7487832696\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-882580a-4913995\right){x}-1344854454a-7487832696$ |
| 9.3-c1 |
9.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.06155021$ |
0.766086219 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 525 a - 2826\) , \( 129 a - 525\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(525a-2826\right){x}+129a-525$ |
| 9.3-c2 |
9.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$1.72349$ |
$(2a+11)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.06155021$ |
0.766086219 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -235 a - 1306\) , \( -1391 a - 7745\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-235a-1306\right){x}-1391a-7745$ |
| 10.1-a1 |
10.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{14} \cdot 5^{4} \) |
$1.76950$ |
$(7a+39), (-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.238859305$ |
$14.75889789$ |
4.432138131 |
\( \frac{168045839}{80000} a - \frac{7265033}{625} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 56167 a - 312681\) , \( -17676050 a + 98416161\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(56167a-312681\right){x}-17676050a+98416161$ |
| 10.1-a2 |
10.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{31}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{7} \cdot 5^{8} \) |
$1.76950$ |
$(7a+39), (-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.477718611$ |
$14.75889789$ |
4.432138131 |
\( -\frac{324776223858953}{6250000} a + \frac{1808366556500473}{6250000} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 907647 a - 5053521\) , \( -1109983170 a + 6180124817\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(907647a-5053521\right){x}-1109983170a+6180124817$ |
*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.
