| 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 |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1$ |
$1.013292176$ |
2.669954154 |
\( -\frac{305464256137}{24} a - \frac{481953441203}{12} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -10 a - 130\) , \( -101 a - 917\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-10a-130\right){x}-101a-917$ |
| 6.1-a2 |
6.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$9.119629590$ |
2.669954154 |
\( -\frac{3411947}{4608} a + \frac{118391}{2304} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}$ |
| 6.1-a3 |
6.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$9.119629590$ |
2.669954154 |
\( -\frac{131406772321033}{402653184} a + \frac{369777222537421}{201326592} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 5 a - 10\) , \( 4 a + 37\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(5a-10\right){x}+4a+37$ |
| 6.1-b1 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$2.089924069$ |
$30.88585248$ |
1.399854627 |
\( -\frac{305464256137}{24} a - \frac{481953441203}{12} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -17525 a - 81029\) , \( 2812460 a + 13003338\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-17525a-81029\right){x}+2812460a+13003338$ |
| 6.1-b2 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.696641356$ |
$30.88585248$ |
1.399854627 |
\( -\frac{3411947}{4608} a + \frac{118391}{2304} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -205 a - 949\) , \( 3788 a + 17514\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-205a-949\right){x}+3788a+17514$ |
| 6.1-b3 |
6.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$2.089924069$ |
$3.431761387$ |
1.399854627 |
\( -\frac{131406772321033}{402653184} a + \frac{369777222537421}{201326592} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 1480 a + 6841\) , \( -25504 a - 117918\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(1480a+6841\right){x}-25504a-117918$ |
| 6.1-c1 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$1.013292176$ |
0.889984718 |
\( -\frac{305464256137}{24} a - \frac{481953441203}{12} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -123620 a - 571549\) , \( -53919510 a - 249295521\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-123620a-571549\right){x}-53919510a-249295521$ |
| 6.1-c2 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$9.119629590$ |
0.889984718 |
\( -\frac{3411947}{4608} a + \frac{118391}{2304} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -1450 a - 6699\) , \( -85399 a - 394834\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-1450a-6699\right){x}-85399a-394834$ |
| 6.1-c3 |
6.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$9.119629590$ |
0.889984718 |
\( -\frac{131406772321033}{402653184} a + \frac{369777222537421}{201326592} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 10435 a + 48251\) , \( 581811 a + 2689995\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(10435a+48251\right){x}+581811a+2689995$ |
| 6.1-d1 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.115433659$ |
$30.88585248$ |
2.087606584 |
\( -\frac{305464256137}{24} a - \frac{481953441203}{12} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 929 a - 5265\) , \( -32129 a + 180702\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(929a-5265\right){x}-32129a+180702$ |
| 6.1-d2 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{21} \cdot 3^{3} \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$0.038477886$ |
$30.88585248$ |
2.087606584 |
\( -\frac{3411947}{4608} a + \frac{118391}{2304} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 9 a - 65\) , \( -a - 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(9a-65\right){x}-a-2$ |
| 6.1-d3 |
6.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{39} \cdot 3 \) |
$1.43308$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{3} \) |
$0.115433659$ |
$3.431761387$ |
2.087606584 |
\( -\frac{131406772321033}{402653184} a + \frac{369777222537421}{201326592} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 374 a - 2115\) , \( 10279 a - 57810\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(374a-2115\right){x}+10279a-57810$ |
*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.
