| 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 |
| 47.1-a1 |
47.1-a |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{4} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$311.2963543$ |
2.220601643 |
\( -\frac{854106486661303}{9759362} a^{3} + \frac{1309708070168852}{4879681} a^{2} - \frac{107768209049327}{4879681} a - \frac{419232771807451}{9759362} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{3}{2}\) , \( -a^{2} + 2\) , \( a^{2} - a - 3\) , \( -\frac{11}{2} a^{3} + 41 a + \frac{31}{2}\) , \( 95 a^{3} - 43 a^{2} - 609 a - 201\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{3}{2}\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-\frac{11}{2}a^{3}+41a+\frac{31}{2}\right){x}+95a^{3}-43a^{2}-609a-201$ |
| 47.1-a2 |
47.1-a |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1245.185417$ |
2.220601643 |
\( -\frac{14995779}{4418} a^{3} + \frac{11982345}{2209} a^{2} + \frac{22108274}{2209} a + \frac{55891815}{4418} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{1}{2}\) , \( a^{2} - a - 3\) , \( a\) , \( -6 a^{3} + 9 a^{2} + 33 a - 15\) , \( -6 a^{3} + 9 a^{2} + 34 a - 19\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{1}{2}\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-6a^{3}+9a^{2}+33a-15\right){x}-6a^{3}+9a^{2}+34a-19$ |
| 47.1-a3 |
47.1-a |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$622.5927087$ |
2.220601643 |
\( \frac{1733458109}{94} a^{3} - \frac{1164969528}{47} a^{2} - \frac{4799335457}{47} a + \frac{5037175485}{94} \) |
\( \bigl[a\) , \( \frac{1}{2} a^{3} - a^{2} - 3 a + \frac{3}{2}\) , \( a^{2} - 2\) , \( \frac{17}{2} a^{3} - 5 a^{2} - 55 a - \frac{33}{2}\) , \( -17 a^{3} + 8 a^{2} + 105 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-a^{2}-3a+\frac{3}{2}\right){x}^{2}+\left(\frac{17}{2}a^{3}-5a^{2}-55a-\frac{33}{2}\right){x}-17a^{3}+8a^{2}+105a+34$ |
| 47.1-a4 |
47.1-a |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$622.5927087$ |
2.220601643 |
\( -\frac{5753380329}{94} a^{3} + \frac{1473018244}{47} a^{2} + \frac{17978945643}{47} a + \frac{11792101919}{94} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( -\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{3}{2}\) , \( -\frac{3}{2} a^{3} + 2 a^{2} + 8 a - \frac{5}{2}\) , \( -\frac{3}{2} a^{3} + 5 a^{2} - 2 a - \frac{1}{2}\bigr] \) |
${y}^2+{x}{y}+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{3}{2}\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-\frac{3}{2}a^{3}+2a^{2}+8a-\frac{5}{2}\right){x}-\frac{3}{2}a^{3}+5a^{2}-2a-\frac{1}{2}$ |
| 47.1-b1 |
47.1-b |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{4} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$442.3767078$ |
3.155650333 |
\( \frac{1453216775921384890493283}{4879681} a^{3} - \frac{1953342392250408271604268}{4879681} a^{2} - \frac{8047056442543374755402460}{4879681} a + \frac{4222617136311468914616572}{4879681} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( -a^{3} + a^{2} + 6 a\) , \( \frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( \frac{605}{2} a^{3} - 160 a^{2} - 1893 a - \frac{1239}{2}\) , \( -5374 a^{3} + 2761 a^{2} + 33596 a + 11016\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a\right){x}^{2}+\left(\frac{605}{2}a^{3}-160a^{2}-1893a-\frac{1239}{2}\right){x}-5374a^{3}+2761a^{2}+33596a+11016$ |
| 47.1-b2 |
47.1-b |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$884.7534156$ |
3.155650333 |
\( \frac{10396814922808000}{47} a^{3} + \frac{19813247048187918}{47} a^{2} - \frac{4809467368048720}{47} a - \frac{3578071461053471}{47} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{1}{2}\) , \( -\frac{1}{2} a^{3} + 3 a + \frac{3}{2}\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a - \frac{1}{2}\) , \( \frac{141}{2} a^{3} - 36 a^{2} - 442 a - \frac{291}{2}\) , \( -\frac{651}{2} a^{3} + 166 a^{2} + 2036 a + \frac{1335}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{1}{2}\right){x}{y}+\left(-\frac{1}{2}a^{3}+a^{2}+2a-\frac{1}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+3a+\frac{3}{2}\right){x}^{2}+\left(\frac{141}{2}a^{3}-36a^{2}-442a-\frac{291}{2}\right){x}-\frac{651}{2}a^{3}+166a^{2}+2036a+\frac{1335}{2}$ |
| 47.1-b3 |
47.1-b |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$884.7534156$ |
3.155650333 |
\( -\frac{1209607162149525}{47} a^{3} + \frac{619405527522758}{47} a^{2} + \frac{7559868467694644}{47} a + \frac{2479067102596890}{47} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{3} - a^{2} - 2 a + \frac{3}{2}\) , \( a\) , \( -\frac{7}{2} a^{3} - 2 a^{2} + 9 a - \frac{9}{2}\) , \( \frac{15}{2} a^{3} + 11 a^{2} - 8 a + \frac{1}{2}\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(\frac{1}{2}a^{3}-a^{2}-2a+\frac{3}{2}\right){x}^{2}+\left(-\frac{7}{2}a^{3}-2a^{2}+9a-\frac{9}{2}\right){x}+\frac{15}{2}a^{3}+11a^{2}-8a+\frac{1}{2}$ |
| 47.1-b4 |
47.1-b |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1769.506831$ |
3.155650333 |
\( \frac{561671912500}{2209} a^{3} - \frac{741986957344}{2209} a^{2} - \frac{3079953887744}{2209} a + \frac{1639520290065}{2209} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{3} + 4 a + \frac{5}{2}\) , \( a^{2} - a - 2\) , \( 10 a^{3} - 23 a^{2} - 18 a - 2\) , \( 11 a^{3} - 39 a^{2} + 16 a + 10\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+4a+\frac{5}{2}\right){x}^{2}+\left(10a^{3}-23a^{2}-18a-2\right){x}+11a^{3}-39a^{2}+16a+10$ |
| 47.1-c1 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$122.6360895$ |
1.749624743 |
\( -\frac{23492516568527}{94} a^{3} + \frac{236890947384951}{47} a^{2} - \frac{685295657774263}{47} a + \frac{558684150026963}{94} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 2 a - 4\) , \( a + 1\) , \( 2 a^{3} - 4 a^{2} + 12 a\) , \( -\frac{7}{2} a^{3} + 20 a^{2} + 4 a - \frac{7}{2}\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(2a^{3}-4a^{2}+12a\right){x}-\frac{7}{2}a^{3}+20a^{2}+4a-\frac{7}{2}$ |
| 47.1-c2 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$981.0887166$ |
1.749624743 |
\( \frac{1491164587}{4418} a^{3} + \frac{273178879}{2209} a^{2} - \frac{8648506911}{2209} a + \frac{9073350511}{4418} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 2 a - 4\) , \( a + 1\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a + \frac{5}{2}\) , \( -a^{3} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-\frac{1}{2}a^{3}+a^{2}+2a+\frac{5}{2}\right){x}-a^{3}+4a+1$ |
| 47.1-c3 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{4} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$981.0887166$ |
1.749624743 |
\( \frac{18449526293}{9759362} a^{3} + \frac{16367807741}{4879681} a^{2} - \frac{9713545769}{4879681} a + \frac{16553556071}{9759362} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a - \frac{3}{2}\) , \( 0\) , \( -\frac{3}{2} a^{3} + 2 a^{2} + 8 a - \frac{7}{2}\) , \( 0\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+a^{2}+2a-\frac{3}{2}\right){x}^{2}+\left(-\frac{3}{2}a^{3}+2a^{2}+8a-\frac{7}{2}\right){x}$ |
| 47.1-c4 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{8} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$61.31804478$ |
1.749624743 |
\( -\frac{3702129450912996533}{47622573323522} a^{3} + \frac{1005004366696757389}{23811286661761} a^{2} + \frac{11414703304637633765}{23811286661761} a + \frac{7518781694793177387}{47622573323522} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a - \frac{3}{2}\) , \( 0\) , \( 6 a^{3} - 8 a^{2} - 32 a + 14\) , \( -\frac{17}{2} a^{3} + 11 a^{2} + 51 a - \frac{63}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+a^{2}+2a-\frac{3}{2}\right){x}^{2}+\left(6a^{3}-8a^{2}-32a+14\right){x}-\frac{17}{2}a^{3}+11a^{2}+51a-\frac{63}{2}$ |
| 47.1-c5 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$490.5443583$ |
1.749624743 |
\( -\frac{822374392417}{94} a^{3} + \frac{210561407929}{47} a^{2} + \frac{2569869914167}{47} a + \frac{1685446537501}{94} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( 1\) , \( \frac{1}{2} a^{3} - 3 a - \frac{1}{2}\) , \( \frac{3}{2} a^{3} - a^{2} - 9 a - \frac{9}{2}\) , \( -a^{3} + a^{2} + 5 a - 1\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-3a-\frac{1}{2}\right){y}={x}^{3}+{x}^{2}+\left(\frac{3}{2}a^{3}-a^{2}-9a-\frac{9}{2}\right){x}-a^{3}+a^{2}+5a-1$ |
| 47.1-c6 |
47.1-c |
$6$ |
$8$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$981.0887166$ |
1.749624743 |
\( \frac{553891020693}{4418} a^{3} + \frac{527720529927}{2209} a^{2} - \frac{128198203669}{2209} a - \frac{190504219731}{4418} \) |
\( \bigl[\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\) , \( a^{2} - 3\) , \( a^{2} - a - 2\) , \( -2 a^{3} + 7 a^{2} + 7 a - 7\) , \( 3 a^{3} + 2 a^{2} - 27 a + 11\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-2a-\frac{3}{2}\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-2a^{3}+7a^{2}+7a-7\right){x}+3a^{3}+2a^{2}-27a+11$ |
| 47.1-d1 |
47.1-d |
$2$ |
$2$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.045329863$ |
$3932.389012$ |
2.543123825 |
\( -\frac{2551637}{94} a^{3} + \frac{3930074}{47} a^{2} - \frac{490056}{47} a - \frac{217423}{94} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a - \frac{3}{2}\) , \( a^{2} - a - 1\) , \( -\frac{1}{2} a^{3} + a^{2} + 2 a - \frac{1}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}+\left(-\frac{1}{2}a^{3}+a^{2}+2a-\frac{3}{2}\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a-1\right){x}^{2}+\left(a^{2}-a-1\right){x}-\frac{1}{2}a^{3}+a^{2}+2a-\frac{1}{2}$ |
| 47.1-d2 |
47.1-d |
$2$ |
$2$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.090659727$ |
$983.0972532$ |
2.543123825 |
\( -\frac{6202943}{4418} a^{3} + \frac{2057850}{2209} a^{2} + \frac{18291612}{2209} a + \frac{12294693}{4418} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{1}{2}\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{1}{2}\) , \( \frac{3}{2} a^{3} - 4 a + \frac{1}{2}\) , \( a^{3} + 2 a^{2} - 2 a\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{1}{2}\right){x}{y}+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{1}{2}\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(\frac{3}{2}a^{3}-4a+\frac{1}{2}\right){x}+a^{3}+2a^{2}-2a$ |
| 47.1-e1 |
47.1-e |
$2$ |
$2$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$227.7727856$ |
1.624794556 |
\( -\frac{290192386}{2209} a^{3} + \frac{862002310}{2209} a^{2} - \frac{14923922}{2209} a - \frac{115274517}{2209} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( -\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{3}{2}\) , \( a^{2} + a - 1\) , \( -a - 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{3}{2}\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(a^{2}+a-1\right){x}-a-2$ |
| 47.1-e2 |
47.1-e |
$2$ |
$2$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$455.5455712$ |
1.624794556 |
\( -\frac{4219664573}{47} a^{3} + \frac{2160892322}{47} a^{2} + \frac{26372558648}{47} a + \frac{8648283502}{47} \) |
\( \bigl[\frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( -a^{3} + a^{2} + 5 a\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 11 a - 1\) , \( a^{3} - a^{2} - 6 a + 1\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+11a-1\right){x}+a^{3}-a^{2}-6a+1$ |
| 47.1-f1 |
47.1-f |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{4} \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.320005781$ |
$447.3595841$ |
2.042401820 |
\( -\frac{70269018055}{9759362} a^{3} + \frac{96200802830}{4879681} a^{2} + \frac{9117142189}{4879681} a - \frac{26991830707}{9759362} \) |
\( \bigl[\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\) , \( -\frac{1}{2} a^{3} + 3 a + \frac{3}{2}\) , \( a + 1\) , \( -2 a^{3} + a^{2} + 12 a + 1\) , \( \frac{1}{2} a^{3} - a^{2} - 3 a + \frac{5}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-2a-\frac{3}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+3a+\frac{3}{2}\right){x}^{2}+\left(-2a^{3}+a^{2}+12a+1\right){x}+\frac{1}{2}a^{3}-a^{2}-3a+\frac{5}{2}$ |
| 47.1-f2 |
47.1-f |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.320005781$ |
$447.3595841$ |
2.042401820 |
\( \frac{76920497940631}{94} a^{3} - \frac{51696371742102}{47} a^{2} - \frac{212970149626957}{47} a + \frac{223508163435527}{94} \) |
\( \bigl[\frac{1}{2} a^{3} - 2 a - \frac{1}{2}\) , \( \frac{1}{2} a^{3} - 3 a - \frac{5}{2}\) , \( 1\) , \( a^{3} + 2 a^{2} - 4 a - 1\) , \( \frac{77}{2} a^{3} - 18 a^{2} - 235 a - \frac{155}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-2a-\frac{1}{2}\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{3}-3a-\frac{5}{2}\right){x}^{2}+\left(a^{3}+2a^{2}-4a-1\right){x}+\frac{77}{2}a^{3}-18a^{2}-235a-\frac{155}{2}$ |
| 47.1-f3 |
47.1-f |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.640011563$ |
$447.3595841$ |
2.042401820 |
\( \frac{9506604517}{4418} a^{3} + \frac{6123495493}{2209} a^{2} - \frac{6708969066}{2209} a + \frac{2904717535}{4418} \) |
\( \bigl[a\) , \( -\frac{1}{2} a^{3} + 2 a + \frac{5}{2}\) , \( \frac{1}{2} a^{3} - 2 a - \frac{1}{2}\) , \( -\frac{7}{2} a^{3} + 2 a^{2} + 20 a - \frac{11}{2}\) , \( \frac{5}{2} a^{3} - 6 a^{2} - 11 a + \frac{15}{2}\bigr] \) |
${y}^2+a{x}{y}+\left(\frac{1}{2}a^{3}-2a-\frac{1}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+2a+\frac{5}{2}\right){x}^{2}+\left(-\frac{7}{2}a^{3}+2a^{2}+20a-\frac{11}{2}\right){x}+\frac{5}{2}a^{3}-6a^{2}-11a+\frac{15}{2}$ |
| 47.1-f4 |
47.1-f |
$4$ |
$4$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47 \) |
$10.13501$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1.280023126$ |
$27.95997400$ |
2.042401820 |
\( \frac{11172350123404541}{94} a^{3} + \frac{10645753519322836}{47} a^{2} - \frac{2583728538594821}{47} a - \frac{3844746905561363}{94} \) |
\( \bigl[a\) , \( -\frac{1}{2} a^{3} + 2 a + \frac{5}{2}\) , \( \frac{1}{2} a^{3} - 2 a - \frac{1}{2}\) , \( -a^{3} - 23 a^{2} + 25 a - 3\) , \( 26 a^{3} - 151 a^{2} + 14 a + 26\bigr] \) |
${y}^2+a{x}{y}+\left(\frac{1}{2}a^{3}-2a-\frac{1}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+2a+\frac{5}{2}\right){x}^{2}+\left(-a^{3}-23a^{2}+25a-3\right){x}+26a^{3}-151a^{2}+14a+26$ |
*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.
