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 |
15.1-a1 |
15.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.551490810$ |
2.040131471 |
\( \frac{5929741}{5625} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -30 a + 261\) , \( 162 a - 979\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-30a+261\right){x}+162a-979$ |
15.1-a2 |
15.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$26.20596324$ |
2.040131471 |
\( \frac{205379}{75} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 15 a - 54\) , \( 63 a - 313\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(15a-54\right){x}+63a-313$ |
15.1-b1 |
15.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$6.551490810$ |
4.080262942 |
\( \frac{5929741}{5625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -7 a + 43\) , \( -27 a + 197\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-7a+43\right){x}-27a+197$ |
15.1-b2 |
15.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$26.20596324$ |
4.080262942 |
\( \frac{205379}{75} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2 a + 8\) , \( 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2a+8\right){x}+9$ |
15.1-c1 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$26.31897150$ |
$0.490422220$ |
2.009680769 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
15.1-c2 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.644935719$ |
$31.38702211$ |
2.009680769 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
15.1-c3 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$13.15948575$ |
$1.961688882$ |
2.009680769 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
15.1-c4 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$6.579742876$ |
$7.846755528$ |
2.009680769 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
15.1-c5 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.289871438$ |
$31.38702211$ |
2.009680769 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
15.1-c6 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$13.15948575$ |
$1.961688882$ |
2.009680769 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
15.1-c7 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.644935719$ |
$31.38702211$ |
2.009680769 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
15.1-c8 |
15.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$26.31897150$ |
$0.490422220$ |
2.009680769 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
15.1-d1 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{44} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.471890948$ |
$2.547989231$ |
2.995347692 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -990\) , \( 22765\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-990{x}+22765$ |
15.1-d2 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.887563793$ |
$10.19195692$ |
2.995347692 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 0\) , \( -5\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-5$ |
15.1-d3 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.775127587$ |
$2.547989231$ |
2.995347692 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 315\) , \( 1066\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+315{x}+1066$ |
15.1-d4 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.887563793$ |
$10.19195692$ |
2.995347692 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -90\) , \( 175\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-90{x}+175$ |
15.1-d5 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.775127587$ |
$10.19195692$ |
2.995347692 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -45\) , \( -104\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-45{x}-104$ |
15.1-d6 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{28} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.943781896$ |
$10.19195692$ |
2.995347692 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1215\) , \( 16600\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-1215{x}+16600$ |
15.1-d7 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$7.550255175$ |
$2.547989231$ |
2.995347692 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -720\) , \( -7259\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-720{x}-7259$ |
15.1-d8 |
15.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.471890948$ |
$10.19195692$ |
2.995347692 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -19440\) , \( 1048135\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-19440{x}+1048135$ |
15.1-e1 |
15.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$6.551490810$ |
4.080262942 |
\( \frac{5929741}{5625} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -634 a + 4385\) , \( 17532 a - 121383\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-634a+4385\right){x}+17532a-121383$ |
15.1-e2 |
15.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$26.20596324$ |
4.080262942 |
\( \frac{205379}{75} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 206 a - 1430\) , \( 2503 a - 17343\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(206a-1430\right){x}+2503a-17343$ |
15.1-f1 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{44} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$9.352879577$ |
$0.490422220$ |
2.856692515 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 12867 a - 89067\) , \( 3850257 a - 26653866\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(12867a-89067\right){x}+3850257a-26653866$ |
15.1-f2 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.584554973$ |
$31.38702211$ |
2.856692515 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -3 a + 33\) , \( -843 a + 5844\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+33\right){x}-843a+5844$ |
15.1-f3 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.676439788$ |
$1.961688882$ |
2.856692515 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -4098 a + 28383\) , \( 170895 a - 1183029\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4098a+28383\right){x}+170895a-1183029$ |
15.1-f4 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.338219894$ |
$7.846755528$ |
2.856692515 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 1167 a - 8067\) , \( 31737 a - 219696\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1167a-8067\right){x}+31737a-219696$ |
15.1-f5 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.169109947$ |
$31.38702211$ |
2.856692515 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 582 a - 4017\) , \( -16305 a + 112881\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(582a-4017\right){x}-16305a+112881$ |
15.1-f6 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{28} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.676439788$ |
$1.961688882$ |
2.856692515 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 15792 a - 109317\) , \( 2820387 a - 19524471\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15792a-109317\right){x}+2820387a-19524471$ |
15.1-f7 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.584554973$ |
$31.38702211$ |
2.856692515 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 9357 a - 64767\) , \( -1200795 a + 8312646\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9357a-64767\right){x}-1200795a+8312646$ |
15.1-f8 |
15.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$9.352879577$ |
$0.490422220$ |
2.856692515 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 252717 a - 1749567\) , \( 176592117 a - 1222480176\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(252717a-1749567\right){x}+176592117a-1222480176$ |
15.1-g1 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$6.664295827$ |
$2.547989231$ |
2.643868672 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 1438 a - 9852\) , \( -139212 a + 963920\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1438a-9852\right){x}-139212a+963920$ |
15.1-g2 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$6.664295827$ |
$10.19195692$ |
2.643868672 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 8 a + 48\) , \( 48 a - 120\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8a+48\right){x}+48a-120$ |
15.1-g3 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.833036978$ |
$2.547989231$ |
2.643868672 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -447 a + 3198\) , \( -7386 a + 51344\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-447a+3198\right){x}-7386a+51344$ |
15.1-g4 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.666073956$ |
$10.19195692$ |
2.643868672 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 138 a - 852\) , \( -852 a + 6110\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(138a-852\right){x}-852a+6110$ |
15.1-g5 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.332147913$ |
$10.19195692$ |
2.643868672 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 73 a - 402\) , \( 774 a - 5146\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(73a-402\right){x}+774a-5146$ |
15.1-g6 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.332147913$ |
$10.19195692$ |
2.643868672 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 1763 a - 12102\) , \( -100302 a + 694560\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1763a-12102\right){x}-100302a+694560$ |
15.1-g7 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$6.664295827$ |
$2.547989231$ |
2.643868672 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 1048 a - 7152\) , \( 46944 a - 324766\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1048a-7152\right){x}+46944a-324766$ |
15.1-g8 |
15.1-g |
$8$ |
$16$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.666073956$ |
$10.19195692$ |
2.643868672 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 28088 a - 194352\) , \( -6474192 a + 44818500\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(28088a-194352\right){x}-6474192a+44818500$ |
15.1-h1 |
15.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{8} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.551490810$ |
2.040131471 |
\( \frac{5929741}{5625} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -5702 a + 39469\) , \( -473385 a + 3277047\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5702a+39469\right){x}-473385a+3277047$ |
15.1-h2 |
15.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{165}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{4} \) |
$2.25893$ |
$(3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$26.20596324$ |
2.040131471 |
\( \frac{205379}{75} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 1858 a - 12866\) , \( -67602 a + 467967\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(1858a-12866\right){x}-67602a+467967$ |
*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.