| 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 |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.93294$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$16.92336820$ |
1.620964690 |
\( -585 a - 1243 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -58 a - 245\) , \( -630 a - 2944\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-58a-245\right){x}-630a-2944$ |
| 1.1-a2 |
1.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.93294$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$16.92336820$ |
1.620964690 |
\( 585 a - 1828 \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 58 a - 304\) , \( 688 a - 3878\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(58a-304\right){x}+688a-3878$ |
| 7.1-a1 |
7.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{6} \) |
$1.51749$ |
$(-a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.996438388$ |
0.430851258 |
\( -\frac{14644619289}{117649} a + \frac{83818782450}{117649} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 131 a - 752\) , \( 1841 a - 10539\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(131a-752\right){x}+1841a-10539$ |
| 7.1-a2 |
7.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{3} \) |
$1.51749$ |
$(-a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.99287677$ |
0.430851258 |
\( \frac{200475}{343} a + \frac{756729}{343} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 6 a - 37\) , \( 40 a - 237\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(6a-37\right){x}+40a-237$ |
| 7.2-a1 |
7.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{3} \) |
$1.51749$ |
$(-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.99287677$ |
0.430851258 |
\( -\frac{200475}{343} a + \frac{957204}{343} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -7 a - 31\) , \( -41 a - 197\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-7a-31\right){x}-41a-197$ |
| 7.2-a2 |
7.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{6} \) |
$1.51749$ |
$(-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.996438388$ |
0.430851258 |
\( \frac{14644619289}{117649} a + \frac{69174163161}{117649} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -132 a - 621\) , \( -1842 a - 8698\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-132a-621\right){x}-1842a-8698$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.61589$ |
$(a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.12147851$ |
3.279880432 |
\( -8019 a - 37638 \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -5322 a - 25083\) , \( 482954 a + 2279657\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-5322a-25083\right){x}+482954a+2279657$ |
| 9.2-b1 |
9.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{3} \) |
$1.61589$ |
$(a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.214474431$ |
1.573607906 |
\( -8019 a - 37638 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -a - 5\) , \( -a - 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-a-5\right){x}-a-5$ |
| 9.2-c1 |
9.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.61589$ |
$(a-6)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.649594216$ |
$26.66883312$ |
1.872774003 |
\( -585 a - 1243 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -107 a + 618\) , \( -715 a + 4102\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-107a+618\right){x}-715a+4102$ |
| 9.2-c2 |
9.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.61589$ |
$(a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.549864738$ |
$8.889611043$ |
1.872774003 |
\( 585 a - 1828 \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 9\) , \( a\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+9{x}+a$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$1.61589$ |
$(a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$17.12147851$ |
3.279880432 |
\( 8019 a - 45657 \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 5321 a - 30405\) , \( -482954 a + 2762611\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(5321a-30405\right){x}-482954a+2762611$ |
| 9.3-b1 |
9.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{3} \) |
$1.61589$ |
$(a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$8.214474431$ |
1.573607906 |
\( 8019 a - 45657 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( a - 6\) , \( a - 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(a-6\right){x}+a-6$ |
| 9.3-c1 |
9.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.61589$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.549864738$ |
$8.889611043$ |
1.872774003 |
\( -585 a - 1243 \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 9\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+9{x}-a+1$ |
| 9.3-c2 |
9.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.61589$ |
$(a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.649594216$ |
$26.66883312$ |
1.872774003 |
\( 585 a - 1828 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 105 a + 511\) , \( 714 a + 3387\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(105a+511\right){x}+714a+3387$ |
| 12.1-a1 |
12.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{36} \cdot 3^{5} \) |
$1.73639$ |
$(a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$0.401742023$ |
3.463191633 |
\( -\frac{7012904377690871}{31850496} a - \frac{66203887751429717}{63700992} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -2926 a + 16736\) , \( 111936 a - 640300\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2926a+16736\right){x}+111936a-640300$ |
| 12.1-a2 |
12.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{12} \cdot 3^{15} \) |
$1.73639$ |
$(a-6), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$3.615678215$ |
3.463191633 |
\( \frac{169986953807}{918330048} a - \frac{813551200609}{459165024} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 379 a - 2164\) , \( -10536 a + 60269\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(379a-2164\right){x}-10536a+60269$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{8} \) |
$1.73639$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.235998508$ |
$14.03330491$ |
2.537733176 |
\( -\frac{955076}{6561} a + \frac{20525879}{26244} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 6\) , \( 3 a + 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+6{x}+3a+9$ |
| 12.1-c1 |
12.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{5} \) |
$1.73639$ |
$(a-6), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1.611741384$ |
$17.01278132$ |
2.101103349 |
\( -\frac{1903561}{972} a - \frac{4496185}{486} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 8 a + 28\) , \( 20 a + 56\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(8a+28\right){x}+20a+56$ |
| 12.1-c2 |
12.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3 \) |
$1.73639$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$8.058706921$ |
$0.680511252$ |
2.101103349 |
\( \frac{1133806928930279}{3072} a - \frac{6485549394369605}{3072} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 703 a - 3947\) , \( 22525 a - 128674\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(703a-3947\right){x}+22525a-128674$ |
| 12.1-d1 |
12.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{3} \) |
$1.73639$ |
$(a-6), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$15.22118499$ |
1.457925107 |
\( -\frac{10909}{216} a - \frac{69241}{108} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 527 a - 3015\) , \( 34652 a - 198215\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(527a-3015\right){x}+34652a-198215$ |
| 12.1-d2 |
12.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3 \) |
$1.73639$ |
$(a-6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.691242776$ |
1.457925107 |
\( \frac{66066912553}{768} a - \frac{755824701137}{1536} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 56647 a - 324030\) , \( 16772306 a - 95940161\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(56647a-324030\right){x}+16772306a-95940161$ |
| 12.2-a1 |
12.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{12} \cdot 3^{15} \) |
$1.73639$ |
$(a+5), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$3.615678215$ |
3.463191633 |
\( -\frac{169986953807}{918330048} a - \frac{53967238793}{34012224} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -379 a - 1785\) , \( 10536 a + 49733\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-379a-1785\right){x}+10536a+49733$ |
| 12.2-a2 |
12.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{36} \cdot 3^{5} \) |
$1.73639$ |
$(a+5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$0.401742023$ |
3.463191633 |
\( \frac{7012904377690871}{31850496} a - \frac{2971470240993017}{2359296} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 2926 a + 13810\) , \( -111936 a - 528364\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(2926a+13810\right){x}-111936a-528364$ |
| 12.2-b1 |
12.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{8} \) |
$1.73639$ |
$(a+5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.235998508$ |
$14.03330491$ |
2.537733176 |
\( \frac{955076}{6561} a + \frac{618725}{972} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( a + 5\) , \( -5 a + 7\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}-5a+7$ |
| 12.2-c1 |
12.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3 \) |
$1.73639$ |
$(a+5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$8.058706921$ |
$0.680511252$ |
2.101103349 |
\( -\frac{1133806928930279}{3072} a - \frac{891957077573221}{512} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -691 a - 3270\) , \( -25783 a - 121704\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-691a-3270\right){x}-25783a-121704$ |
| 12.2-c2 |
12.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{5} \) |
$1.73639$ |
$(a+5), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1.611741384$ |
$17.01278132$ |
2.101103349 |
\( \frac{1903561}{972} a - \frac{403553}{36} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 4 a + 10\) , \( 2 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4a+10\right){x}+2a+6$ |
| 12.2-d1 |
12.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3 \) |
$1.73639$ |
$(a+5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.691242776$ |
1.457925107 |
\( -\frac{66066912553}{768} a - \frac{207896958677}{512} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -56647 a - 267383\) , \( -16772306 a - 79167855\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-56647a-267383\right){x}-16772306a-79167855$ |
| 12.2-d2 |
12.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{3} \) |
$1.73639$ |
$(a+5), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$15.22118499$ |
1.457925107 |
\( \frac{10909}{216} a - \frac{5533}{8} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -527 a - 2488\) , \( -34652 a - 163563\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-527a-2488\right){x}-34652a-163563$ |
| 15.1-a1 |
15.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{7} \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.067446852$ |
$20.21196446$ |
1.828037073 |
\( \frac{3457390514176}{2109375} a + \frac{16315221118976}{2109375} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 256 a - 1457\) , \( -7467 a + 42708\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(256a-1457\right){x}-7467a+42708$ |
| 15.1-b1 |
15.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.431085428$ |
$21.57202858$ |
1.781439496 |
\( \frac{1095737135104}{135} a - \frac{6267779485696}{135} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 51 a - 292\) , \( -471 a + 2686\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(51a-292\right){x}-471a+2686$ |
| 15.1-b2 |
15.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.143695142$ |
$21.57202858$ |
1.781439496 |
\( \frac{20342665216}{2460375} a + \frac{86936662016}{2460375} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -99 a - 467\) , \( 1245 a + 5871\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-99a-467\right){x}+1245a+5871$ |
| 15.1-c1 |
15.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{3} \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.580589682$ |
$23.48290995$ |
3.869602557 |
\( \frac{446464}{3375} a + \frac{3313664}{3375} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -a + 6\) , \( -3 a + 9\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-a+6\right){x}-3a+9$ |
| 15.1-c2 |
15.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{9} \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.741769048$ |
$2.609212217$ |
3.869602557 |
\( \frac{58724238094336}{5859375} a + \frac{277184943067136}{5859375} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 9 a - 54\) , \( 82 a - 480\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(9a-54\right){x}+82a-480$ |
| 15.1-d1 |
15.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.83601$ |
$(a-6), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.454975188$ |
$12.07073453$ |
3.364387672 |
\( \frac{77824}{15} a + \frac{241664}{15} \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( -5 a - 19\) , \( -6 a - 32\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a-19\right){x}-6a-32$ |
| 15.2-a1 |
15.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{3} \cdot 5^{6} \) |
$1.83601$ |
$(a-6), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.227634701$ |
$9.484510107$ |
2.481540506 |
\( \frac{10308743}{421875} a - \frac{256851386}{421875} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -314 a - 1451\) , \( -14903 a - 70315\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-314a-1451\right){x}-14903a-70315$ |
| 15.2-a2 |
15.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{9} \cdot 5^{2} \) |
$1.83601$ |
$(a-6), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.682904103$ |
$9.484510107$ |
2.481540506 |
\( -\frac{7716517}{492075} a + \frac{237223159}{492075} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -197 a + 1158\) , \( 4822 a - 27522\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-197a+1158\right){x}+4822a-27522$ |
| 15.3-a1 |
15.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.3 |
\( 3 \cdot 5 \) |
\( 3^{3} \cdot 5^{6} \) |
$1.83601$ |
$(a+5), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.227634701$ |
$9.484510107$ |
2.481540506 |
\( -\frac{10308743}{421875} a - \frac{9131209}{15625} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 314 a - 1765\) , \( 15217 a - 86983\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(314a-1765\right){x}+15217a-86983$ |
| 15.3-a2 |
15.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.3 |
\( 3 \cdot 5 \) |
\( 3^{9} \cdot 5^{2} \) |
$1.83601$ |
$(a+5), (3a+14)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.682904103$ |
$9.484510107$ |
2.481540506 |
\( \frac{7716517}{492075} a + \frac{8500246}{18225} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 197 a + 961\) , \( -5019 a - 23661\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(197a+961\right){x}-5019a-23661$ |
| 15.4-a1 |
15.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{7} \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.067446852$ |
$20.21196446$ |
1.828037073 |
\( -\frac{3457390514176}{2109375} a + \frac{732318949376}{78125} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -256 a - 1201\) , \( 7466 a + 35241\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-256a-1201\right){x}+7466a+35241$ |
| 15.4-b1 |
15.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.431085428$ |
$21.57202858$ |
1.781439496 |
\( -\frac{1095737135104}{135} a - \frac{191557124096}{5} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -51 a - 241\) , \( 470 a + 2215\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-51a-241\right){x}+470a+2215$ |
| 15.4-b2 |
15.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.143695142$ |
$21.57202858$ |
1.781439496 |
\( -\frac{20342665216}{2460375} a + \frac{3973308416}{91125} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 99 a - 566\) , \( -1246 a + 7116\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(99a-566\right){x}-1246a+7116$ |
| 15.4-c1 |
15.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{9} \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.741769048$ |
$2.609212217$ |
3.869602557 |
\( -\frac{58724238094336}{5859375} a + \frac{111969727053824}{1953125} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -9 a - 45\) , \( -83 a - 398\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-9a-45\right){x}-83a-398$ |
| 15.4-c2 |
15.4-c |
$2$ |
$3$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{3} \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.580589682$ |
$23.48290995$ |
3.869602557 |
\( -\frac{446464}{3375} a + \frac{139264}{125} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( a + 5\) , \( 2 a + 6\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(a+5\right){x}+2a+6$ |
| 15.4-d1 |
15.4-d |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
15.4 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.83601$ |
$(a+5), (3a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.454975188$ |
$12.07073453$ |
3.364387672 |
\( -\frac{77824}{15} a + \frac{106496}{5} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( 7 a - 25\) , \( 11 a - 63\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a-25\right){x}+11a-63$ |
| 16.1-a1 |
16.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$1.86587$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$5.922289431$ |
0.567252448 |
\( -16 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -2175 a - 10266\) , \( -1317789 a - 6220166\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-2175a-10266\right){x}-1317789a-6220166$ |
| 21.1-a1 |
21.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7 \) |
$1.99713$ |
$(a+5), (-a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$47.53019671$ |
4.552567175 |
\( -\frac{756386879}{21} a - \frac{1190095568}{7} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -15 a - 65\) , \( 33 a + 157\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-15a-65\right){x}+33a+157$ |
| 21.1-b1 |
21.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{3} \cdot 7 \) |
$1.99713$ |
$(a+5), (-a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$14.07572796$ |
4.044630669 |
\( -\frac{71564}{189} a - \frac{12511}{7} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 3 a - 11\) , \( -2335 a + 13353\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a-11\right){x}-2335a+13353$ |
| 21.2-a1 |
21.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$1.99713$ |
$(a-6), (-a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.113177823$ |
$11.34187913$ |
0.983610348 |
\( \frac{542764456}{35721} a - \frac{3267781225}{35721} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -a\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}-a{x}$ |
| 21.2-b1 |
21.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{14} \) |
$1.99713$ |
$(a-6), (-a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1.020053663$ |
$3.422829345$ |
2.340954958 |
\( -\frac{34066465075516948}{18312022966923} a + \frac{184226600824936171}{18312022966923} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 147901 a - 846002\) , \( 67871043 a - 388232757\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(147901a-846002\right){x}+67871043a-388232757$ |
| 21.2-b2 |
21.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{109}) \) |
$2$ |
$[2, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7^{7} \) |
$1.99713$ |
$(a-6), (-a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.510026831$ |
$6.845658691$ |
2.340954958 |
\( \frac{79043995497128}{600362847} a + \frac{373119932678719}{600362847} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -6094 a + 34873\) , \( 4182709 a - 23925726\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6094a+34873\right){x}+4182709a-23925726$ |
*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.
