| 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 |
| 6.1-a1 |
6.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$13.52926920$ |
3.131417342 |
\( -\frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 361 a - 2327\) , \( -7786 a + 50451\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(361a-2327\right){x}-7786a+50451$ |
| 6.1-a2 |
6.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$13.52926920$ |
3.131417342 |
\( -\frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 81 a - 512\) , \( 1192 a - 7734\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(81a-512\right){x}+1192a-7734$ |
| 6.1-a3 |
6.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$1.503252134$ |
3.131417342 |
\( -\frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 6571 a - 42572\) , \( 752882 a - 4879242\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(6571a-42572\right){x}+752882a-4879242$ |
| 6.1-b1 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{3} \) |
$0.470597059$ |
$1.751695888$ |
3.434369660 |
\( \frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 15 a - 108\) , \( 220 a - 1443\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-108\right){x}+220a-1443$ |
| 6.1-b2 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$0.156865686$ |
$15.76526299$ |
3.434369660 |
\( \frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -5 a + 27\) , \( -8 a + 57\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5a+27\right){x}-8a+57$ |
| 6.1-b3 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.470597059$ |
$15.76526299$ |
3.434369660 |
\( \frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -15 a - 33\) , \( -8 a + 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a-33\right){x}-8a+249$ |
| 6.1-c1 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$9.351598892$ |
$1.751695888$ |
2.527667452 |
\( -\frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -72 a - 486\) , \( -1492 a - 9762\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-72a-486\right){x}-1492a-9762$ |
| 6.1-c2 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$3.117199630$ |
$15.76526299$ |
2.527667452 |
\( -\frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 8 a + 54\) , \( 12 a + 78\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a+54\right){x}+12a+78$ |
| 6.1-c3 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$9.351598892$ |
$15.76526299$ |
2.527667452 |
\( -\frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 48 a - 186\) , \( -148 a + 2574\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(48a-186\right){x}-148a+2574$ |
| 6.1-d1 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$13.52926920$ |
1.043805780 |
\( \frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -1451 a - 9341\) , \( 67294 a + 436248\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1451a-9341\right){x}+67294a+436248$ |
| 6.1-d2 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$13.52926920$ |
1.043805780 |
\( \frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -331 a - 2081\) , \( -8430 a - 54492\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-331a-2081\right){x}-8430a-54492$ |
| 6.1-d3 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$1.503252134$ |
1.043805780 |
\( \frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -26291 a - 170321\) , \( -5931590 a - 38440956\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-26291a-170321\right){x}-5931590a-38440956$ |
| 6.1-e1 |
6.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$13.52926920$ |
3.131417342 |
\( \frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -361 a - 2327\) , \( 7786 a + 50451\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-361a-2327\right){x}+7786a+50451$ |
| 6.1-e2 |
6.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$13.52926920$ |
3.131417342 |
\( \frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -81 a - 512\) , \( -1192 a - 7734\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-81a-512\right){x}-1192a-7734$ |
| 6.1-e3 |
6.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$1.503252134$ |
3.131417342 |
\( \frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -6571 a - 42572\) , \( -752882 a - 4879242\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-6571a-42572\right){x}-752882a-4879242$ |
| 6.1-f1 |
6.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{3} \) |
$0.470597059$ |
$1.751695888$ |
3.434369660 |
\( -\frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -17 a - 108\) , \( -221 a - 1443\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-17a-108\right){x}-221a-1443$ |
| 6.1-f2 |
6.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$0.156865686$ |
$15.76526299$ |
3.434369660 |
\( -\frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 3 a + 27\) , \( 7 a + 57\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(3a+27\right){x}+7a+57$ |
| 6.1-f3 |
6.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.470597059$ |
$15.76526299$ |
3.434369660 |
\( -\frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 13 a - 33\) , \( 7 a + 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(13a-33\right){x}+7a+249$ |
| 6.1-g1 |
6.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$9.351598892$ |
$1.751695888$ |
2.527667452 |
\( \frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -973785 a - 6310804\) , \( -1206878846 a - 7821468799\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-973785a-6310804\right){x}-1206878846a-7821468799$ |
| 6.1-g2 |
6.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$3.117199630$ |
$15.76526299$ |
2.527667452 |
\( \frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -218825 a - 1418104\) , \( 140749150 a + 912158801\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-218825a-1418104\right){x}+140749150a+912158801$ |
| 6.1-g3 |
6.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$9.351598892$ |
$15.76526299$ |
2.527667452 |
\( \frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -17715825 a - 114811624\) , \( 103259966350 a + 669201066497\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-17715825a-114811624\right){x}+103259966350a+669201066497$ |
| 6.1-h1 |
6.1-h |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$13.52926920$ |
1.043805780 |
\( -\frac{76452242473}{49152} a - \frac{82500763787}{8192} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -55140 a - 357339\) , \( 17768356 a + 115152129\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-55140a-357339\right){x}+17768356a+115152129$ |
| 6.1-h2 |
6.1-h |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$13.52926920$ |
1.043805780 |
\( -\frac{10513825}{288} a + \frac{3785397}{16} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -100 a - 639\) , \( 64500 a + 418029\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-100a-639\right){x}+64500a+418029$ |
| 6.1-h3 |
6.1-h |
$3$ |
$9$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.81272$ |
$(2,a), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$1.503252134$ |
1.043805780 |
\( -\frac{97620665254362025}{12} a + \frac{105442369719932349}{2} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 900 a + 5841\) , \( -1746500 a - 11318595\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(900a+5841\right){x}-1746500a-11318595$ |
| 9.1-a1 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{26} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$16.45138346$ |
$2.246907049$ |
5.703781587 |
\( -\frac{873722816}{59049} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 126 a - 747\) , \( 2090 a - 13389\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(126a-747\right){x}+2090a-13389$ |
| 9.1-a2 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$3.290276692$ |
$11.23453524$ |
5.703781587 |
\( \frac{64}{9} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 6 a + 33\) , \( 5 a + 135\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(6a+33\right){x}+5a+135$ |
| 9.1-a3 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1.645138346$ |
$22.46907049$ |
5.703781587 |
\( \frac{85184}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 22 a - 129\) , \( -185 a + 1208\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(22a-129\right){x}-185a+1208$ |
| 9.1-a4 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{16} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$8.225691731$ |
$4.493814098$ |
5.703781587 |
\( \frac{58591911104}{243} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1942 a - 12609\) , \( 114655 a - 743032\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(1942a-12609\right){x}+114655a-743032$ |
| 9.1-b1 |
9.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{26} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$4.427518132$ |
$2.246907049$ |
1.535043934 |
\( -\frac{873722816}{59049} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -478 a - 3093\) , \( -16373 a - 106088\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-478a-3093\right){x}-16373a-106088$ |
| 9.1-b2 |
9.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{10} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$0.885503626$ |
$11.23453524$ |
1.535043934 |
\( \frac{64}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 2 a + 27\) , \( 67 a + 424\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(2a+27\right){x}+67a+424$ |
| 9.1-b3 |
9.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$0.442751813$ |
$22.46907049$ |
1.535043934 |
\( \frac{85184}{3} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( a + 15\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a+15\right){x}+2$ |
| 9.1-b4 |
9.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.213759066$ |
$4.493814098$ |
1.535043934 |
\( \frac{58591911104}{243} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -479 a - 3105\) , \( -17595 a - 114028\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-479a-3105\right){x}-17595a-114028$ |
| 9.1-c1 |
9.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{26} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$4.427518132$ |
$2.246907049$ |
1.535043934 |
\( -\frac{873722816}{59049} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -322650 a - 2090997\) , \( 269081541 a + 1743847704\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-322650a-2090997\right){x}+269081541a+1743847704$ |
| 9.1-c2 |
9.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{10} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$0.885503626$ |
$11.23453524$ |
1.535043934 |
\( \frac{64}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1350 a + 8763\) , \( -1072179 a - 6948504\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(1350a+8763\right){x}-1072179a-6948504$ |
| 9.1-c3 |
9.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$0.442751813$ |
$22.46907049$ |
1.535043934 |
\( \frac{85184}{3} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -3720 a - 24009\) , \( -308755 a - 2000780\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-3720a-24009\right){x}-308755a-2000780$ |
| 9.1-c4 |
9.1-c |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.213759066$ |
$4.493814098$ |
1.535043934 |
\( \frac{58591911104}{243} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -327720 a - 2123769\) , \( 259525040 a + 1681914670\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-327720a-2123769\right){x}+259525040a+1681914670$ |
| 9.1-d1 |
9.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{26} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$16.45138346$ |
$2.246907049$ |
5.703781587 |
\( -\frac{873722816}{59049} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -127 a - 747\) , \( -2091 a - 13389\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-127a-747\right){x}-2091a-13389$ |
| 9.1-d2 |
9.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$3.290276692$ |
$11.23453524$ |
5.703781587 |
\( \frac{64}{9} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -7 a + 33\) , \( -6 a + 135\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-7a+33\right){x}-6a+135$ |
| 9.1-d3 |
9.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1.645138346$ |
$22.46907049$ |
5.703781587 |
\( \frac{85184}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -22 a - 129\) , \( 185 a + 1208\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-22a-129\right){x}+185a+1208$ |
| 9.1-d4 |
9.1-d |
$4$ |
$10$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{16} \) |
$2.00611$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$8.225691731$ |
$4.493814098$ |
5.703781587 |
\( \frac{58591911104}{243} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -1942 a - 12609\) , \( -114655 a - 743032\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-1942a-12609\right){x}-114655a-743032$ |
| 14.1-a1 |
14.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{3} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$24.21346718$ |
3.736219100 |
\( \frac{9930171131}{392} a - \frac{9193553137}{56} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 48 a - 204\) , \( -272 a + 1976\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(48a-204\right){x}-272a+1976$ |
| 14.1-a2 |
14.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{14} \cdot 7 \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$24.21346718$ |
3.736219100 |
\( \frac{225}{14} a - \frac{169}{2} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 8 a + 56\) , \( 16 a + 112\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a+56\right){x}+16a+112$ |
| 14.1-b1 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{48} \cdot 7^{2} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -35476 a - 229804\) , \( 9223680 a + 59776488\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-35476a-229804\right){x}+9223680a+59776488$ |
| 14.1-b2 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{16} \cdot 7^{2} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -116 a - 644\) , \( -3360 a - 21560\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-116a-644\right){x}-3360a-21560$ |
| 14.1-b3 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{24} \cdot 7^{6} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 924 a + 6096\) , \( 67680 a + 438832\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(924a+6096\right){x}+67680a+438832$ |
| 14.1-b4 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{12} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -7396 a - 47824\) , \( 708960 a + 4594800\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7396a-47824\right){x}+708960a+4594800$ |
| 14.1-b5 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{14} \cdot 7^{4} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -2196 a - 14124\) , \( -145440 a - 942344\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2196a-14124\right){x}-145440a-942344$ |
| 14.1-b6 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{30} \cdot 7^{4} \) |
$2.24040$ |
$(2,a), (a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$7.027708105$ |
1.084398903 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -567956 a - 3680684\) , \( 592166400 a + 3837677032\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-567956a-3680684\right){x}+592166400a+3837677032$ |
| 14.1-c1 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$2.24040$ |
$(2,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.484059793$ |
$7.027708105$ |
3.778110619 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -8871 a - 57416\) , \( 1135220 a + 7357190\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8871a-57416\right){x}+1135220a+7357190$ |
| 14.1-c2 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$2.24040$ |
$(2,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.484059793$ |
$7.027708105$ |
3.778110619 |
\( -\frac{15625}{28} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -31 a - 126\) , \( -480 a - 2986\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-31a-126\right){x}-480a-2986$ |
*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.
