| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$1.95731$ |
$(-a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.246034057$ |
$36.36107212$ |
2.150067107 |
\( -\frac{233}{9} a + \frac{2027}{9} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -10 a + 90\) , \( -29 a + 290\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+90\right){x}-29a+290$ |
| 3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{2} \) |
$1.95731$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.246034057$ |
$36.36107212$ |
2.150067107 |
\( \frac{233}{9} a + \frac{598}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 8 a + 81\) , \( 28 a + 262\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(8a+81\right){x}+28a+262$ |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{2} \) |
$2.10326$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$25.64077500$ |
1.540604858 |
\( \frac{130301}{2} a + \frac{1015855}{2} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 72\) , \( -a + 100\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+72{x}-a+100$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{2} \) |
$2.10326$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$25.64077500$ |
1.540604858 |
\( -\frac{130301}{2} a + 573078 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -a + 72\) , \( a + 99\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a+72\right){x}+a+99$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{4} \) |
$2.57596$ |
$(-a+9), (a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$11.95994771$ |
2.155810839 |
\( \frac{120912137}{27} a + \frac{311848732}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13004 a - 101691\) , \( 2225941 a + 17410597\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13004a-101691\right){x}+2225941a+17410597$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{4} \) |
$2.57596$ |
$(-a+9), (a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$11.95994771$ |
2.155810839 |
\( -\frac{120912137}{27} a + \frac{1056458333}{27} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 13002 a - 114693\) , \( -2225942 a + 19636539\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13002a-114693\right){x}-2225942a+19636539$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{8} \) |
$2.57596$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$2.289646042$ |
$11.55947745$ |
6.361018504 |
\( \frac{233}{9} a + \frac{598}{3} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 148 a - 965\) , \( 52510 a - 461797\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(148a-965\right){x}+52510a-461797$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{8} \) |
$2.57596$ |
$(-a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$2.289646042$ |
$11.55947745$ |
6.361018504 |
\( -\frac{233}{9} a + \frac{2027}{9} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -112 a - 888\) , \( -53363 a - 417387\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-112a-888\right){x}-53363a-417387$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{26} \cdot 3^{2} \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.04209797$ |
1.326910734 |
\( \frac{15729213152957}{73728} a + \frac{30615130265371}{18432} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -166 a - 1334\) , \( 3340 a + 26192\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-166a-1334\right){x}+3340a+26192$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{14} \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$9$ |
\( 2 \) |
$1$ |
$3.784625123$ |
4.093129532 |
\( \frac{2763070771481}{9565938} a - \frac{23943622896461}{9565938} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -320942 a - 2510291\) , \( 276374663 a + 2161708233\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-320942a-2510291\right){x}+276374663a+2161708233$ |
| 12.1-c1 |
12.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{3} \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$26.38845886$ |
1.585528828 |
\( -\frac{37021}{54} a - \frac{133528}{27} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -7133 a + 62921\) , \( 1674559 a - 14772407\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-7133a+62921\right){x}+1674559a-14772407$ |
| 12.1-c2 |
12.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3 \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$26.38845886$ |
1.585528828 |
\( \frac{1609}{12} a + \frac{13063}{24} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 608 a + 4760\) , \( -276 a - 2174\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(608a+4760\right){x}-276a-2174$ |
| 12.1-d1 |
12.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.76805$ |
$(-a+9), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.386041549$ |
$12.34848474$ |
4.582767427 |
\( -\frac{202847}{324} a - \frac{864203}{162} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -30155 a - 235794\) , \( -9039448 a - 70703367\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-30155a-235794\right){x}-9039448a-70703367$ |
| 12.1-e1 |
12.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{14} \cdot 3^{24} \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$1.577089494$ |
0.189516248 |
\( \frac{3661340597238869}{36150980669568} a + \frac{30366968102020375}{36150980669568} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -10659469 a + 94034229\) , \( 323902938247 a - 2857361104614\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-10659469a+94034229\right){x}+323902938247a-2857361104614$ |
| 12.1-f1 |
12.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.76805$ |
$(-a+9), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 2^{2} \) |
$1$ |
$0.537777700$ |
0.516990886 |
\( \frac{50621998071145}{144} a - \frac{223285120276703}{72} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -4389 a - 34325\) , \( -458549 a - 3586629\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-4389a-34325\right){x}-458549a-3586629$ |
| 12.2-a1 |
12.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{26} \cdot 3^{2} \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.04209797$ |
1.326910734 |
\( -\frac{15729213152957}{73728} a + \frac{46063244738147}{24576} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 165 a - 1499\) , \( -3341 a + 29533\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(165a-1499\right){x}-3341a+29533$ |
| 12.2-b1 |
12.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{14} \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$9$ |
\( 2 \) |
$1$ |
$3.784625123$ |
4.093129532 |
\( -\frac{2763070771481}{9565938} a - \frac{3530092020830}{1594323} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 320979 a - 2831305\) , \( -279205967 a + 2463060526\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(320979a-2831305\right){x}-279205967a+2463060526$ |
| 12.2-c1 |
12.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3 \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$26.38845886$ |
1.585528828 |
\( -\frac{1609}{12} a + \frac{5427}{8} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -609 a + 5368\) , \( 275 a - 2450\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-609a+5368\right){x}+275a-2450$ |
| 12.2-c2 |
12.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{3} \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$26.38845886$ |
1.585528828 |
\( \frac{37021}{54} a - \frac{101359}{18} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 7132 a + 55788\) , \( -1674560 a - 13097848\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(7132a+55788\right){x}-1674560a-13097848$ |
| 12.2-d1 |
12.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.76805$ |
$(a+8), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.386041549$ |
$12.34848474$ |
4.582767427 |
\( \frac{202847}{324} a - \frac{643751}{108} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 30155 a - 265949\) , \( 9039448 a - 79742815\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(30155a-265949\right){x}+9039448a-79742815$ |
| 12.2-e1 |
12.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{14} \cdot 3^{24} \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$1.577089494$ |
0.189516248 |
\( -\frac{3661340597238869}{36150980669568} a + \frac{2835692391604937}{3012581722464} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10659471 a + 83374758\) , \( -323913597717 a - 2533541541126\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10659471a+83374758\right){x}-323913597717a-2533541541126$ |
| 12.2-f1 |
12.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.76805$ |
$(a+8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 2^{2} \) |
$1$ |
$0.537777700$ |
0.516990886 |
\( -\frac{50621998071145}{144} a - \frac{131982747494087}{48} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 4388 a - 38714\) , \( 458548 a - 4045178\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(4388a-38714\right){x}+458548a-4045178$ |
| 13.1-a1 |
13.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$2.82400$ |
$(a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$2.768702153$ |
$13.89497266$ |
4.623001627 |
\( \frac{2391}{13} a + \frac{1007}{13} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 57 a + 424\) , \( 436 a + 3397\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(57a+424\right){x}+436a+3397$ |
| 13.2-a1 |
13.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{277}) \) |
$2$ |
$[2, 0]$ |
13.2 |
\( 13 \) |
\( 13 \) |
$2.82400$ |
$(a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$2.768702153$ |
$13.89497266$ |
4.623001627 |
\( -\frac{2391}{13} a + \frac{3398}{13} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -23 a + 412\) , \( 10 a + 644\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-23a+412\right){x}+10a+644$ |
*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.
