| 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 |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{18} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$25.18055127$ |
1.365607129 |
\( -\frac{505895}{729} a + \frac{3864485}{729} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 7 a + 6\) , \( 9 a - 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(7a+6\right){x}+9a-1$ |
| 3.1-a2 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{24} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$6.295137819$ |
1.365607129 |
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -3 a + 21\) , \( -190\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-3a+21\right){x}-190$ |
| 3.1-a3 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{14} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$2.797839030$ |
1.365607129 |
\( -\frac{392415680105}{9} a + \frac{2007752749790}{9} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 177 a - 1059\) , \( 2964 a - 16948\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(177a-1059\right){x}+2964a-16948$ |
| 3.1-a4 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{16} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2 \) |
$1$ |
$0.699459757$ |
1.365607129 |
\( \frac{732096671152080845}{81} a + \frac{3008750567734015615}{81} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -148 a - 2394\) , \( -5717 a - 52648\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-148a-2394\right){x}-5717a-52648$ |
| 3.1-b1 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.357504263$ |
$25.18055127$ |
1.235878333 |
\( -\frac{505895}{729} a + \frac{3864485}{729} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+a+2$ |
| 3.1-b2 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.715008527$ |
$6.295137819$ |
1.235878333 |
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -29 a - 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-4a-19\right){x}-29a-121$ |
| 3.1-b3 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.072512791$ |
$2.797839030$ |
1.235878333 |
\( -\frac{392415680105}{9} a + \frac{2007752749790}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 119\) , \( -245 a - 1051\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24a-119\right){x}-245a-1051$ |
| 3.1-b4 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{4} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$8.145025583$ |
$0.699459757$ |
1.235878333 |
\( \frac{732096671152080845}{81} a + \frac{3008750567734015615}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -434 a - 1804\) , \( -12042 a - 49534\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-434a-1804\right){x}-12042a-49534$ |
| 3.2-a1 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{16} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2 \) |
$1$ |
$0.699459757$ |
1.365607129 |
\( -\frac{732096671152080845}{81} a + \frac{1246949079628698820}{27} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10332 a - 42522\) , \( -1271183 a - 5224453\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-10332a-42522\right){x}-1271183a-5224453$ |
| 3.2-a2 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{24} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$6.295137819$ |
1.365607129 |
\( -\frac{25207270205}{531441} a + \frac{42974373355}{177147} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 198 a + 813\) , \( -7340 a - 30166\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(198a+813\right){x}-7340a-30166$ |
| 3.2-a3 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{18} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$25.18055127$ |
1.365607129 |
\( \frac{505895}{729} a + \frac{1119530}{243} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -127 a - 522\) , \( -2077 a - 8536\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-127a-522\right){x}-2077a-8536$ |
| 3.2-a4 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{14} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$2.797839030$ |
1.365607129 |
\( \frac{392415680105}{9} a + \frac{538445689895}{3} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10342 a - 42507\) , \( -1270240 a - 5220400\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-10342a-42507\right){x}-1270240a-5220400$ |
| 3.2-b1 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{4} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$8.145025583$ |
$0.699459757$ |
1.235878333 |
\( -\frac{732096671152080845}{81} a + \frac{1246949079628698820}{27} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13050 a - 53636\) , \( -1785556 a - 7338231\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13050a-53636\right){x}-1785556a-7338231$ |
| 3.2-b2 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{12} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.715008527$ |
$6.295137819$ |
1.235878333 |
\( -\frac{25207270205}{531441} a + \frac{42974373355}{177147} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 245 a + 1004\) , \( -10714 a - 44034\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(245a+1004\right){x}-10714a-44034$ |
| 3.2-b3 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.357504263$ |
$25.18055127$ |
1.235878333 |
\( \frac{505895}{729} a + \frac{1119530}{243} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -165 a - 681\) , \( -2743 a - 11275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-165a-681\right){x}-2743a-11275$ |
| 3.2-b4 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{2} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.072512791$ |
$2.797839030$ |
1.235878333 |
\( \frac{392415680105}{9} a + \frac{538445689895}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13055 a - 53656\) , \( -1784186 a - 7332600\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13055a-53656\right){x}-1784186a-7332600$ |
| 4.1-a1 |
4.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \cdot 3^{12} \) |
$1.16510$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.3 |
$25$ |
\( 1 \) |
$1$ |
$0.769953241$ |
2.087828865 |
\( -\frac{18170704189}{32} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -383 a - 1644\) , \( -9842 a - 40758\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-383a-1644\right){x}-9842a-40758$ |
| 4.1-a2 |
4.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \cdot 3^{12} \) |
$1.16510$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.4 |
$1$ |
\( 1 \) |
$1$ |
$19.24883104$ |
2.087828865 |
\( \frac{1331}{2} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 2 a + 6\) , \( a + 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(2a+6\right){x}+a+3$ |
| 4.1-b1 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.16510$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$0.769953241$ |
0.083513154 |
\( -\frac{18170704189}{32} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -490 a - 2009\) , \( -14487 a - 59538\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-490a-2009\right){x}-14487a-59538$ |
| 4.1-b2 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$1.16510$ |
$(2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$19.24883104$ |
0.083513154 |
\( \frac{1331}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 5 a + 26\) , \( 7 a + 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(5a+26\right){x}+7a+30$ |
| 9.2-a1 |
9.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$3.130085864$ |
$7.265400118$ |
2.466643153 |
\( -12534349815 a + 64047678245 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 6 a - 26\) , \( 24 a - 113\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a-26\right){x}+24a-113$ |
| 9.2-a2 |
9.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.565042932$ |
$14.53080023$ |
2.466643153 |
\( -35685 a + 194030 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 4\) , \( -5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+4\right){x}-5$ |
| 9.2-a3 |
9.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.521680977$ |
$43.59240071$ |
2.466643153 |
\( 35685 a + 158345 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -177 a - 716\) , \( 2528 a + 10395\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-177a-716\right){x}+2528a+10395$ |
| 9.2-a4 |
9.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.043361954$ |
$21.79620035$ |
2.466643153 |
\( 12534349815 a + 51513328430 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -2812 a - 11546\) , \( 171148 a + 703383\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2812a-11546\right){x}+171148a+703383$ |
| 9.2-b1 |
9.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.075744777$ |
$7.265400118$ |
1.635776737 |
\( -12534349815 a + 64047678245 \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -3722 a - 15290\) , \( 215742 a + 886647\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3722a-15290\right){x}+215742a+886647$ |
| 9.2-b2 |
9.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1.037872388$ |
$14.53080023$ |
1.635776737 |
\( -35685 a + 194030 \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -1087 a - 4460\) , \( -40234 a - 165357\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1087a-4460\right){x}-40234a-165357$ |
| 9.2-b3 |
9.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$3.113617165$ |
$43.59240071$ |
1.635776737 |
\( 35685 a + 158345 \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -222 a - 905\) , \( 3530 a + 14503\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-222a-905\right){x}+3530a+14503$ |
| 9.2-b4 |
9.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$6.227234331$ |
$21.79620035$ |
1.635776737 |
\( 12534349815 a + 51513328430 \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -3547 a - 14570\) , \( 241714 a + 993385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3547a-14570\right){x}+241714a+993385$ |
| 9.3-a1 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.043361954$ |
$21.79620035$ |
2.466643153 |
\( -12534349815 a + 64047678245 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 46 a - 294\) , \( -473 a + 2288\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(46a-294\right){x}-473a+2288$ |
| 9.3-a2 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.521680977$ |
$43.59240071$ |
2.466643153 |
\( -35685 a + 194030 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( a - 24\) , \( -5 a + 47\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(a-24\right){x}-5a+47$ |
| 9.3-a3 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.565042932$ |
$14.53080023$ |
2.466643153 |
\( 35685 a + 158345 \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}$ |
| 9.3-a4 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$3.130085864$ |
$7.265400118$ |
2.466643153 |
\( 12534349815 a + 51513328430 \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -4 a - 20\) , \( -29 a - 109\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-20\right){x}-29a-109$ |
| 9.3-b1 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$6.227234331$ |
$21.79620035$ |
1.635776737 |
\( -12534349815 a + 64047678245 \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -2 a - 21\) , \( -46 a - 191\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a-21\right){x}-46a-191$ |
| 9.3-b2 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$3.113617165$ |
$43.59240071$ |
1.635776737 |
\( -35685 a + 194030 \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 3 a + 4\) , \( 2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+4\right){x}+2a+4$ |
| 9.3-b3 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1.037872388$ |
$14.53080023$ |
1.635776737 |
\( 35685 a + 158345 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -1997 a - 8205\) , \( -103494 a - 425341\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-1997a-8205\right){x}-103494a-425341$ |
| 9.3-b4 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$1.42695$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.075744777$ |
$7.265400118$ |
1.635776737 |
\( 12534349815 a + 51513328430 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -31922 a - 131190\) , \( -6657447 a - 27360595\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-31922a-131190\right){x}-6657447a-27360595$ |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{24} \) |
$1.53336$ |
$(3,a), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.253138682$ |
$9.720342945$ |
1.601333870 |
\( -\frac{2087352478285}{531441} a + \frac{21331791431651}{1062882} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -23 a - 92\) , \( -57847 a - 237737\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-23a-92\right){x}-57847a-237737$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{18} \) |
$1.53336$ |
$(3,a), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.126569341$ |
$19.44068589$ |
1.601333870 |
\( \frac{4133602}{729} a + \frac{87766517}{2916} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -673 a - 2762\) , \( -23395 a - 96149\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-673a-2762\right){x}-23395a-96149$ |
| 12.1-b1 |
12.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{12} \) |
$1.53336$ |
$(3,a), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$9.720342945$ |
2.108638445 |
\( -\frac{2087352478285}{531441} a + \frac{21331791431651}{1062882} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 14 a - 36\) , \( -24 a + 162\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(14a-36\right){x}-24a+162$ |
| 12.1-b2 |
12.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.53336$ |
$(3,a), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.44068589$ |
2.108638445 |
\( \frac{4133602}{729} a + \frac{87766517}{2916} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 4 a + 14\) , \( 4 a + 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(4a+14\right){x}+4a+16$ |
| 12.2-a1 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{18} \) |
$1.53336$ |
$(3,a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.126569341$ |
$19.44068589$ |
1.601333870 |
\( -\frac{4133602}{729} a + \frac{34766975}{972} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -3749 a - 15408\) , \( 223496 a + 918518\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-3749a-15408\right){x}+223496a+918518$ |
| 12.2-a2 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{24} \) |
$1.53336$ |
$(3,a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.253138682$ |
$9.720342945$ |
1.601333870 |
\( \frac{2087352478285}{531441} a + \frac{5719028825027}{354294} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -57679 a - 237048\) , \( 16063422 a + 66017006\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-57679a-237048\right){x}+16063422a+66017006$ |
| 12.2-b1 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.53336$ |
$(3,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.44068589$ |
2.108638445 |
\( -\frac{4133602}{729} a + \frac{34766975}{972} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -3 a - 4\) , \( -a + 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a-4\right){x}-a+3$ |
| 12.2-b2 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{12} \) |
$1.53336$ |
$(3,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$9.720342945$ |
2.108638445 |
\( \frac{2087352478285}{531441} a + \frac{5719028825027}{354294} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -13 a - 44\) , \( 37 a + 161\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-13a-44\right){x}+37a+161$ |
| 17.1-a1 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{8} \) |
$1.67286$ |
$(17,a+8)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.555777033$ |
$2.393455763$ |
2.365420732 |
\( -\frac{35937}{83521} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 9 a - 39\) , \( 1563 a - 7983\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(9a-39\right){x}+1563a-7983$ |
| 17.1-a2 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$1.67286$ |
$(17,a+8)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.138944258$ |
$38.29529222$ |
2.365420732 |
\( \frac{35937}{17} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 9 a - 39\) , \( -5 a + 39\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(9a-39\right){x}-5a+39$ |
| 17.1-a3 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{4} \) |
$1.67286$ |
$(17,a+8)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$4.555777033$ |
$9.573823055$ |
2.365420732 |
\( \frac{20346417}{289} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 64 a - 324\) , \( 608 a - 3108\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(64a-324\right){x}+608a-3108$ |
| 17.1-a4 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$1.67286$ |
$(17,a+8)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.555777033$ |
$2.393455763$ |
2.365420732 |
\( \frac{82483294977}{17} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 999 a - 5169\) , \( 37685 a - 192981\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(999a-5169\right){x}+37685a-192981$ |
| 17.1-b1 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$1.67286$ |
$(17,a+8)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$7.904266758$ |
$2.393455763$ |
2.052000824 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( -14\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$ |
| 17.1-b2 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$1.67286$ |
$(17,a+8)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.976066689$ |
$38.29529222$ |
2.052000824 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$ |
*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.
