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 |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.29185$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$11$ |
11Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.214699673 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 12880 a - 99542\) , \( -2141323 a + 16549035\bigr] \) |
${y}^2+{y}={x}^{3}+\left(12880a-99542\right){x}-2141323a+16549035$ |
1.1-a2 |
1.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.29185$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$11$ |
11Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.214699673 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -12880 a - 86662\) , \( 2141323 a + 14407712\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-12880a-86662\right){x}+2141323a+14407712$ |
2.1-a1 |
2.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.048108188$ |
$34.45290976$ |
2.751585456 |
\( -\frac{1714092597}{4096} a - \frac{11539289943}{4096} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 333486 a - 2577189\) , \( -412096662 a + 3184854795\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(333486a-2577189\right){x}-412096662a+3184854795$ |
2.1-b1 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -166349 a - 1119240\) , \( -101485499 a - 682836650\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-166349a-1119240\right){x}-101485499a-682836650$ |
2.1-b2 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( -\frac{102557}{1024} a + \frac{1072561}{1024} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -6449241 a + 49842606\) , \( -1496864675 a + 11568393744\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6449241a+49842606\right){x}-1496864675a+11568393744$ |
2.1-c1 |
2.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.529462610$ |
$13.03498084$ |
1.909556628 |
\( \frac{2361203}{4} a - \frac{18549919}{4} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 1502 a - 11391\) , \( 84221 a - 650261\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1502a-11391\right){x}+84221a-650261$ |
2.1-c2 |
2.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$0.105892522$ |
$13.03498084$ |
1.909556628 |
\( -\frac{102557}{1024} a + \frac{1072561}{1024} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 8 a + 57\) , \( 19 a + 137\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(8a+57\right){x}+19a+137$ |
2.1-d1 |
2.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$2.835776404$ |
0.392309511 |
\( -\frac{1714092597}{4096} a - \frac{11539289943}{4096} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -1797 a - 12082\) , \( -124118 a - 835100\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1797a-12082\right){x}-124118a-835100$ |
2.2-a1 |
2.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.048108188$ |
$34.45290976$ |
2.751585456 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -333486 a - 2243755\) , \( 412430147 a + 2775001836\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-333486a-2243755\right){x}+412430147a+2775001836$ |
2.2-b1 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 166348 a - 1285588\) , \( 101485499 a - 784322149\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(166348a-1285588\right){x}+101485499a-784322149$ |
2.2-b2 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 6449267 a + 43393366\) , \( 1546707281 a + 10406890251\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6449267a+43393366\right){x}+1546707281a+10406890251$ |
2.2-c1 |
2.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{2} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$0.529462610$ |
$13.03498084$ |
1.909556628 |
\( -\frac{2361203}{4} a - 4047179 \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -1476 a - 9940\) , \( -95638 a - 643494\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1476a-9940\right){x}-95638a-643494$ |
2.2-c2 |
2.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$0.105892522$ |
$13.03498084$ |
1.909556628 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -10 a + 66\) , \( -20 a + 156\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+66\right){x}-20a+156$ |
2.2-d1 |
2.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$2.835776404$ |
0.392309511 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 1797 a - 13879\) , \( 122321 a - 945339\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1797a-13879\right){x}+122321a-945339$ |
4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 3 \) |
$0.299859083$ |
$19.84955375$ |
2.470279326 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -403 a - 2708\) , \( -267565 a - 1800300\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-403a-2708\right){x}-267565a-1800300$ |
4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 3 \) |
$0.299859083$ |
$19.84955375$ |
2.470279326 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 402 a - 3111\) , \( 267564 a - 2067865\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(402a-3111\right){x}+267564a-2067865$ |
4.1-c1 |
4.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{33} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$2.636252383$ |
4.102951284 |
\( -\frac{24345866052441}{1073741824} a - \frac{160359237040419}{1073741824} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 25 a - 196\) , \( -132 a + 1019\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(25a-196\right){x}-132a+1019$ |
4.1-c2 |
4.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{21} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$2.636252383$ |
4.102951284 |
\( -\frac{239886134047017}{32768} a + \frac{463484975124303}{8192} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 109626655 a - 847240411\) , \( 1679777810708 a - 12982021956321\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(109626655a-847240411\right){x}+1679777810708a-12982021956321$ |
4.1-c3 |
4.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{33} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$2.636252383$ |
4.102951284 |
\( \frac{24345866052441}{1073741824} a - \frac{46176275773215}{268435456} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -25 a - 171\) , \( 132 a + 887\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-25a-171\right){x}+132a+887$ |
4.1-c4 |
4.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{21} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$2.636252383$ |
4.102951284 |
\( \frac{239886134047017}{32768} a + \frac{1614053766450195}{32768} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -109626655 a - 737613756\) , \( -1679777810708 a - 11302244145613\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-109626655a-737613756\right){x}-1679777810708a-11302244145613$ |
4.1-d1 |
4.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{33} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \) |
$1$ |
$8.606301134$ |
0.297655148 |
\( -\frac{24345866052441}{1073741824} a - \frac{160359237040419}{1073741824} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1485209680 a - 9993108793\) , \( 83988775088712 a + 565111430507277\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-1485209680a-9993108793\right){x}+83988775088712a+565111430507277$ |
4.1-d2 |
4.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{21} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \) |
$1$ |
$8.606301134$ |
0.297655148 |
\( -\frac{239886134047017}{32768} a + \frac{463484975124303}{8192} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 950 a + 6392\) , \( -42668 a - 287088\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(950a+6392\right){x}-42668a-287088$ |
4.1-d3 |
4.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{33} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \) |
$1$ |
$8.606301134$ |
0.297655148 |
\( \frac{24345866052441}{1073741824} a - \frac{46176275773215}{268435456} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 1485209680 a - 11478318473\) , \( -83988775088712 a + 649100205595989\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(1485209680a-11478318473\right){x}-83988775088712a+649100205595989$ |
4.1-d4 |
4.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{21} \) |
$1.82695$ |
$(11a+74), (11a-85)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3, 5$ |
2B, 3Nn, 5B |
$1$ |
\( 2 \) |
$1$ |
$8.606301134$ |
0.297655148 |
\( \frac{239886134047017}{32768} a + \frac{1614053766450195}{32768} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -950 a + 7342\) , \( 42668 a - 329756\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-950a+7342\right){x}+42668a-329756$ |
4.2-a1 |
4.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.82695$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1.797684321$ |
$5.540110081$ |
1.377807922 |
\( -45056 a - 303104 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( 11\) , \( -a - 11\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+11{x}-a-11$ |
4.2-b1 |
4.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.82695$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.323850701$ |
$36.24219350$ |
4.871216415 |
\( -45056 a - 303104 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -37023561 a + 286133487\) , \( -1880699198318 a + 14534826052669\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-37023561a+286133487\right){x}-1880699198318a+14534826052669$ |
4.3-a1 |
4.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.82695$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1.797684321$ |
$5.540110081$ |
1.377807922 |
\( 45056 a - 348160 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( 2 a + 10\) , \( a - 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+10\right){x}+a-2$ |
4.3-b1 |
4.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.82695$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.323850701$ |
$36.24219350$ |
4.871216415 |
\( 45056 a - 348160 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 37023561 a + 249109926\) , \( 1880699198317 a + 12654126854351\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(37023561a+249109926\right){x}+1880699198317a+12654126854351$ |
10.1-a1 |
10.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$1$ |
$3.072924513$ |
6.801876264 |
\( \frac{20383137018133}{610351562500} a - \frac{49963875368579}{152587890625} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 19297589 a - 149139785\) , \( -251996522818 a + 1947533996049\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(19297589a-149139785\right){x}-251996522818a+1947533996049$ |
10.1-b1 |
10.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.971375992$ |
$16.34187476$ |
2.228425904 |
\( -\frac{56211}{50} a + \frac{217211}{25} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 12491185 a - 96537059\) , \( 61050477245 a - 471823494147\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(12491185a-96537059\right){x}+61050477245a-471823494147$ |
10.1-b2 |
10.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5 \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.985687996$ |
$32.68374952$ |
2.228425904 |
\( \frac{68921}{20} a + \frac{137842}{5} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -a + 17\) , \( -2 a - 8\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+17\right){x}-2a-8$ |
10.1-c1 |
10.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$43.47954356$ |
1.503771458 |
\( -\frac{56211}{50} a + \frac{217211}{25} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 3 a + 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+3a+20$ |
10.1-c2 |
10.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5 \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$43.47954356$ |
1.503771458 |
\( \frac{68921}{20} a + \frac{137842}{5} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 512138585 a - 3958020110\) , \( 13200346700953 a - 102017772594930\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(512138585a-3958020110\right){x}+13200346700953a-102017772594930$ |
10.1-d1 |
10.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$2.29727$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$0.109314649$ |
$2.711568399$ |
1.312217311 |
\( \frac{20383137018133}{610351562500} a - \frac{49963875368579}{152587890625} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -265 a - 1737\) , \( 31126 a + 209482\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-265a-1737\right){x}+31126a+209482$ |
10.4-a1 |
10.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$1$ |
$3.072924513$ |
6.801876264 |
\( -\frac{20383137018133}{610351562500} a - \frac{179472364456183}{610351562500} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -19297590 a - 129842196\) , \( 251996522818 a + 1695537473231\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-19297590a-129842196\right){x}+251996522818a+1695537473231$ |
10.4-b1 |
10.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5 \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.985687996$ |
$32.68374952$ |
2.228425904 |
\( -\frac{68921}{20} a + \frac{620289}{20} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 16\) , \( a - 10\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+16{x}+a-10$ |
10.4-b2 |
10.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.971375992$ |
$16.34187476$ |
2.228425904 |
\( \frac{56211}{50} a + \frac{378211}{50} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -12491186 a - 84045873\) , \( -61050477246 a - 410773016901\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-12491186a-84045873\right){x}-61050477246a-410773016901$ |
10.4-c1 |
10.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5 \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$43.47954356$ |
1.503771458 |
\( -\frac{68921}{20} a + \frac{620289}{20} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -512138585 a - 3445881525\) , \( -13200346700953 a - 88817425893977\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-512138585a-3445881525\right){x}-13200346700953a-88817425893977$ |
10.4-c2 |
10.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$43.47954356$ |
1.503771458 |
\( \frac{56211}{50} a + \frac{378211}{50} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( -3 a + 23\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-3a+23$ |
10.4-d1 |
10.4-d |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$2.29727$ |
$(11a+74), (-4a-27)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{5} \) |
$0.109314649$ |
$2.711568399$ |
1.312217311 |
\( -\frac{20383137018133}{610351562500} a - \frac{179472364456183}{610351562500} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 264 a - 2002\) , \( -31126 a + 240608\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(264a-2002\right){x}-31126a+240608$ |
11.1-a1 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$58.18256016$ |
$0.064435690$ |
1.037304259 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$ |
11.1-a2 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{10} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$11.63651203$ |
$1.610892258$ |
1.037304259 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$ |
11.1-a3 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$2.327302406$ |
$40.27230645$ |
1.037304259 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}$ |
11.1-b1 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$5.612837583$ |
$8.512583687$ |
13.21997756 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 4688891060561 a - 36237701385816\) , \( -14859019779017908422 a + 114836688394705117771\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4688891060561a-36237701385816\right){x}-14859019779017908422a+114836688394705117771$ |
11.1-b2 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{10} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1.122567516$ |
$8.512583687$ |
13.21997756 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -6195627759 a - 41686761846\) , \( 1292706345403712 a + 8697866248261405\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6195627759a-41686761846\right){x}+1292706345403712a+8697866248261405$ |
11.1-b3 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$2.35266$ |
$(70a+471)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$5.612837583$ |
$8.512583687$ |
13.21997756 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 199858961 a - 1544593196\) , \( 9820121401118 a - 75893984805809\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(199858961a-1544593196\right){x}+9820121401118a-75893984805809$ |
16.4-a1 |
16.4-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.404753223$ |
$38.24607426$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( -317 a - 2113\) , \( 7753 a + 52137\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-317a-2113\right){x}+7753a+52137$ |
16.4-a2 |
16.4-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$4.452285453$ |
$3.476915842$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 931337 a - 7197740\) , \( 1351685914 a - 10446391276\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(931337a-7197740\right){x}+1351685914a-10446391276$ |
16.5-a1 |
16.5-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$4.452285453$ |
$3.476915842$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -931337 a - 6266403\) , \( -1351685915 a - 9094705362\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-931337a-6266403\right){x}-1351685915a-9094705362$ |
16.5-a2 |
16.5-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.404753223$ |
$38.24607426$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 317 a - 2430\) , \( -7754 a + 59890\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(317a-2430\right){x}-7754a+59890$ |
*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.