| 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 |
| 14.1-a1 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.208472088$ |
$7.027708105$ |
4.457589942 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -145\) , \( 387\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-145{x}+387$ |
| 14.1-a2 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.876248795$ |
$7.027708105$ |
4.457589942 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 25\) , \( 23\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+25{x}+23$ |
| 14.1-a3 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.625416265$ |
$7.027708105$ |
4.457589942 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 30\) , \( 44\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+30{x}+44$ |
| 14.1-a4 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.250832530$ |
$7.027708105$ |
4.457589942 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -10\) , \( -12\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-10{x}-12$ |
| 14.1-a5 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.752497591$ |
$7.027708105$ |
4.457589942 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 15\) , \( -19\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+15{x}-19$ |
| 14.1-a6 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.416944176$ |
$7.027708105$ |
4.457589942 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -2705\) , \( 46979\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-2705{x}+46979$ |
| 14.1-b1 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$32.38901229$ |
$0.436190660$ |
2.388031465 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 14.1-b2 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$3.598779144$ |
$35.33144352$ |
2.388031465 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 14.1-b3 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$10.79633743$ |
$3.925715946$ |
2.388031465 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 14.1-b4 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$5.398168716$ |
$3.925715946$ |
2.388031465 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 14.1-b5 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$1.799389572$ |
$35.33144352$ |
2.388031465 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 14.1-b6 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$16.19450614$ |
$0.436190660$ |
2.388031465 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
| 14.1-c1 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{48} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$2.054541211$ |
$7.027708105$ |
2.440588440 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -8188 a - 48398\) , \( 966826 a + 5719868\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-8188a-48398\right){x}+966826a+5719868$ |
| 14.1-c2 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{16} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$2.054541211$ |
$7.027708105$ |
2.440588440 |
\( -\frac{15625}{28} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -28 a - 118\) , \( -390 a - 2244\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-28a-118\right){x}-390a-2244$ |
| 14.1-c3 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{24} \cdot 7^{6} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.684847070$ |
$7.027708105$ |
2.440588440 |
\( \frac{9938375}{21952} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 212 a + 1302\) , \( 7434 a + 44044\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(212a+1302\right){x}+7434a+44044$ |
| 14.1-c4 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{12} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.342423535$ |
$7.027708105$ |
2.440588440 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -1708 a - 10058\) , \( 72970 a + 431756\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-1708a-10058\right){x}+72970a+431756$ |
| 14.1-c5 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{14} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.027270605$ |
$7.027708105$ |
2.440588440 |
\( \frac{128787625}{98} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -508 a - 2958\) , \( -16038 a - 94820\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-508a-2958\right){x}-16038a-94820$ |
| 14.1-c6 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{30} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.027270605$ |
$7.027708105$ |
2.440588440 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -131068 a - 775438\) , \( 62562474 a + 370124604\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-131068a-775438\right){x}+62562474a+370124604$ |
| 14.1-d1 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{48} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$3.507207167$ |
$0.436190660$ |
4.654534627 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -8182 a - 48398\) , \( -1015936 a - 6010364\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-8182a-48398\right){x}-1015936a-6010364$ |
| 14.1-d2 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{16} \cdot 7^{2} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.389689685$ |
$35.33144352$ |
4.654534627 |
\( -\frac{15625}{28} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -22 a - 118\) , \( 240 a + 1428\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-22a-118\right){x}+240a+1428$ |
| 14.1-d3 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{24} \cdot 7^{6} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.169069055$ |
$3.925715946$ |
4.654534627 |
\( \frac{9938375}{21952} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 218 a + 1302\) , \( -6144 a - 36340\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(218a+1302\right){x}-6144a-36340$ |
| 14.1-d4 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{12} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.338138111$ |
$3.925715946$ |
4.654534627 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -1702 a - 10058\) , \( -83200 a - 492212\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-1702a-10058\right){x}-83200a-492212$ |
| 14.1-d5 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{14} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.779379370$ |
$35.33144352$ |
4.654534627 |
\( \frac{128787625}{98} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -502 a - 2958\) , \( 13008 a + 76964\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-502a-2958\right){x}+13008a+76964$ |
| 14.1-d6 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{30} \cdot 7^{4} \) |
$2.04519$ |
$(2,a+1), (7,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$7.014414334$ |
$0.436190660$ |
4.654534627 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -131062 a - 775438\) , \( -63348864 a - 374777340\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-131062a-775438\right){x}-63348864a-374777340$ |
| 19.1-a1 |
19.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$2.20745$ |
$(a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$4.134983576$ |
$10.05169650$ |
7.025530673 |
\( \frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 5 a + 15\) , \( -9 a - 62\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5a+15\right){x}-9a-62$ |
| 19.1-a2 |
19.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{5} \) |
$2.20745$ |
$(a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.826996715$ |
$10.05169650$ |
7.025530673 |
\( \frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -825 a - 4895\) , \( 26910 a + 159193\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-825a-4895\right){x}+26910a+159193$ |
| 19.1-b1 |
19.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 2^{12} \cdot 19 \) |
$2.20745$ |
$(a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.4 |
$1$ |
\( 1 \) |
$1$ |
$55.57807397$ |
4.697204568 |
\( \frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a - 13\) , \( -4 a + 24\bigr] \) |
${y}^2={x}^{3}+\left(2a-13\right){x}-4a+24$ |
| 19.1-b2 |
19.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 2^{12} \cdot 19^{5} \) |
$2.20745$ |
$(a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.3 |
$25$ |
\( 1 \) |
$1$ |
$2.223122958$ |
4.697204568 |
\( \frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -38 a - 53\) , \( -116 a - 1096\bigr] \) |
${y}^2={x}^{3}+\left(-38a-53\right){x}-116a-1096$ |
| 19.1-c1 |
19.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$2.20745$ |
$(a+4)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$55.57807397$ |
0.187888182 |
\( \frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 5 a + 15\) , \( 3 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5a+15\right){x}+3a+9$ |
| 19.1-c2 |
19.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{5} \) |
$2.20745$ |
$(a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.223122958$ |
0.187888182 |
\( \frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -825 a - 4895\) , \( -36806 a - 217756\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-825a-4895\right){x}-36806a-217756$ |
| 19.1-d1 |
19.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 2^{12} \cdot 19 \) |
$2.20745$ |
$(a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.969321370$ |
$10.05169650$ |
1.646922386 |
\( \frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a - 13\) , \( 4 a - 24\bigr] \) |
${y}^2={x}^{3}+\left(2a-13\right){x}+4a-24$ |
| 19.1-d2 |
19.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( - 2^{12} \cdot 19^{5} \) |
$2.20745$ |
$(a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.193864274$ |
$10.05169650$ |
1.646922386 |
\( \frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -38 a - 53\) , \( 116 a + 1096\bigr] \) |
${y}^2={x}^{3}+\left(-38a-53\right){x}+116a+1096$ |
| 19.2-a1 |
19.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( -19 \) |
$2.20745$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$4.134983576$ |
$10.05169650$ |
7.025530673 |
\( -\frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 12 a + 32\) , \( 24 a + 69\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(12a+32\right){x}+24a+69$ |
| 19.2-a2 |
19.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 19^{5} \) |
$2.20745$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.826996715$ |
$10.05169650$ |
7.025530673 |
\( -\frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 842 a - 4878\) , \( -31805 a + 188374\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(842a-4878\right){x}-31805a+188374$ |
| 19.2-b1 |
19.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 2^{12} \cdot 19 \) |
$2.20745$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.4 |
$1$ |
\( 1 \) |
$1$ |
$55.57807397$ |
4.697204568 |
\( -\frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 13\) , \( 4 a + 24\bigr] \) |
${y}^2={x}^{3}+\left(-2a-13\right){x}+4a+24$ |
| 19.2-b2 |
19.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 2^{12} \cdot 19^{5} \) |
$2.20745$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.3 |
$25$ |
\( 1 \) |
$1$ |
$2.223122958$ |
4.697204568 |
\( -\frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 38 a - 53\) , \( 116 a - 1096\bigr] \) |
${y}^2={x}^{3}+\left(38a-53\right){x}+116a-1096$ |
| 19.2-c1 |
19.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( -19 \) |
$2.20745$ |
$(a-4)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$55.57807397$ |
0.187888182 |
\( -\frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 12 a + 32\) , \( 12 a + 140\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(12a+32\right){x}+12a+140$ |
| 19.2-c2 |
19.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 19^{5} \) |
$2.20745$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.223122958$ |
0.187888182 |
\( -\frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 842 a - 4878\) , \( 31911 a - 188575\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(842a-4878\right){x}+31911a-188575$ |
| 19.2-d1 |
19.2-d |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 2^{12} \cdot 19 \) |
$2.20745$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.969321370$ |
$10.05169650$ |
1.646922386 |
\( -\frac{1866240}{19} a - \frac{10977984}{19} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 13\) , \( -4 a - 24\bigr] \) |
${y}^2={x}^{3}+\left(-2a-13\right){x}-4a-24$ |
| 19.2-d2 |
19.2-d |
$2$ |
$5$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( - 2^{12} \cdot 19^{5} \) |
$2.20745$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.193864274$ |
$10.05169650$ |
1.646922386 |
\( -\frac{19429567370408448}{2476099} a + \frac{114946874211721536}{2476099} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 38 a - 53\) , \( -116 a + 1096\bigr] \) |
${y}^2={x}^{3}+\left(38a-53\right){x}-116a+1096$ |
| 20.1-a1 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.772687765$ |
1.348375146 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 105 a - 633\) , \( 2716 a - 16072\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(105a-633\right){x}+2716a-16072$ |
| 20.1-a2 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.95418988$ |
1.348375146 |
\( \frac{21296}{25} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -15 a + 77\) , \( -98 a + 576\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-15a+77\right){x}-98a+576$ |
| 20.1-a3 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$31.90837977$ |
1.348375146 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 64 a - 367\) , \( -591 a + 3504\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(64a-367\right){x}-591a+3504$ |
| 20.1-a4 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.545375530$ |
1.348375146 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1984 a - 11727\) , \( 113073 a - 668944\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(1984a-11727\right){x}+113073a-668944$ |
| 20.1-b1 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$10.34365470$ |
2.622595135 |
\( -\frac{20720464}{15625} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 111 a - 633\) , \( -2063 a + 12256\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(111a-633\right){x}-2063a+12256$ |
| 20.1-b2 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$10.34365470$ |
2.622595135 |
\( \frac{21296}{25} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -9 a + 77\) , \( 31 a - 132\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-9a+77\right){x}+31a-132$ |
| 20.1-b3 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$20.68730941$ |
2.622595135 |
\( \frac{16384}{5} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 64 a - 367\) , \( 591 a - 3504\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(64a-367\right){x}+591a-3504$ |
| 20.1-b4 |
20.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{20} \cdot 5^{6} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$20.68730941$ |
2.622595135 |
\( \frac{488095744}{125} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1984 a - 11727\) , \( -113073 a + 668944\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(1984a-11727\right){x}-113073a+668944$ |
| 20.1-c1 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{12} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1.129583454$ |
$10.34365470$ |
2.962440071 |
\( -\frac{20720464}{15625} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -36\) , \( 140\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-36{x}+140$ |
| 20.1-c2 |
20.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{35}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$2.23594$ |
$(2,a+1), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$3.388750362$ |
$10.34365470$ |
2.962440071 |
\( \frac{21296}{25} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 4\) , \( -4\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+4{x}-4$ |
*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.
