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 |
128.2-a1 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{22} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.710189503$ |
1.314635010 |
\( -292271882500 a + 748669862250 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 81 a - 220\) , \( 596 a - 1488\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(81a-220\right){x}+596a-1488$ |
128.2-a2 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{14} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$21.68151603$ |
1.314635010 |
\( -20500 a + 54000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 16 a + 25\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a+25\right){x}$ |
128.2-a3 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{16} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$21.68151603$ |
1.314635010 |
\( 2000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+a{x}$ |
128.2-a4 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{14} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$10.84075801$ |
1.314635010 |
\( 20500 a + 33500 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -14 a + 40\) , \( 15 a - 40\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-14a+40\right){x}+15a-40$ |
128.2-a5 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{20} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.84075801$ |
1.314635010 |
\( 1098500 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 20\) , \( 12 a - 16\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-20\right){x}+12a-16$ |
128.2-a6 |
128.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{22} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.420379007$ |
1.314635010 |
\( 292271882500 a + 456397979750 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -79 a - 140\) , \( 516 a + 752\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-79a-140\right){x}+516a+752$ |
128.2-b1 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{16} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$23.36225846$ |
1.416544989 |
\( -7659605 a + 19620476 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 28 a - 71\) , \( -142 a + 364\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(28a-71\right){x}-142a+364$ |
128.2-b2 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{23} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.840564617$ |
1.416544989 |
\( 343 a + 686 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 37 a + 60\) , \( -38 a - 60\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(37a+60\right){x}-38a-60$ |
128.2-b3 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{22} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$11.68112923$ |
1.416544989 |
\( -1995 a + 7016 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a - 7\) , \( 2 a - 6\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(3a-7\right){x}+2a-6$ |
128.2-b4 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{20} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$23.36225846$ |
1.416544989 |
\( 24225 a + 44228 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -33 a - 51\) , \( 178 a + 278\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-33a-51\right){x}+178a+278$ |
128.2-b5 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{23} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$2.920282308$ |
1.416544989 |
\( -21069823 a + 54821634 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 43 a - 127\) , \( 218 a - 606\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(43a-127\right){x}+218a-606$ |
128.2-b6 |
128.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{22} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$11.68112923$ |
1.416544989 |
\( 2701312025 a + 4218241728 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -533 a - 831\) , \( 10078 a + 15738\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-533a-831\right){x}+10078a+15738$ |
128.2-c1 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{15} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$2.324603913$ |
$6.204671930$ |
1.749094730 |
\( -774198 a + 1983150 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 9 a - 23\) , \( 21 a - 54\bigr] \) |
${y}^2={x}^{3}+\left(9a-23\right){x}+21a-54$ |
128.2-c2 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.162301956$ |
$12.40934386$ |
1.749094730 |
\( -349618194 a + 895566564 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 192 a - 491\) , \( -2076 a + 5318\bigr] \) |
${y}^2={x}^{3}+\left(192a-491\right){x}-2076a+5318$ |
128.2-c3 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{18} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.581150978$ |
$24.81868772$ |
1.749094730 |
\( -8748 a + 25056 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 12 a - 31\) , \( -32 a + 82\bigr] \) |
${y}^2={x}^{3}+\left(12a-31\right){x}-32a+82$ |
128.2-c4 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( 2^{18} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.162301956$ |
$12.40934386$ |
1.749094730 |
\( 8748 a + 16308 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -12 a - 19\) , \( -32 a - 50\bigr] \) |
${y}^2={x}^{3}+\left(-12a-19\right){x}-32a-50$ |
128.2-c5 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{15} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.290575489$ |
$24.81868772$ |
1.749094730 |
\( 774198 a + 1208952 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -9 a - 14\) , \( 21 a + 33\bigr] \) |
${y}^2={x}^{3}+\left(-9a-14\right){x}+21a+33$ |
128.2-c6 |
128.2-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.2 |
\( 2^{7} \) |
\( - 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$2.324603913$ |
$3.102335965$ |
1.749094730 |
\( 349618194 a + 545948370 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -192 a - 299\) , \( -2076 a - 3242\bigr] \) |
${y}^2={x}^{3}+\left(-192a-299\right){x}-2076a-3242$ |
*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.