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 |
135.2-a1 |
135.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.140502805$ |
2.438354577 |
\( \frac{336226}{135} a - \frac{1062059}{135} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 8\) , \( -22 a - 69\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+8{x}-22a-69$ |
135.2-a2 |
135.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.140502805$ |
2.438354577 |
\( -\frac{156415766633}{3645} a + \frac{98945821613}{729} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -55 a - 167\) , \( -336 a - 1064\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-55a-167\right){x}-336a-1064$ |
135.2-b1 |
135.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 2^{12} \cdot 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.459288555$ |
$16.24433986$ |
2.359324574 |
\( \frac{1026304}{3645} a + \frac{949568}{729} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -4 a - 10\) , \( -10 a - 32\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-10\right){x}-10a-32$ |
135.2-b2 |
135.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{10} \cdot 5^{6} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.688932833$ |
$3.609853302$ |
2.359324574 |
\( -\frac{66006784}{375} a + \frac{226222784}{375} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -16 a - 62\) , \( -113 a - 373\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-16a-62\right){x}-113a-373$ |
135.2-b3 |
135.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.229644277$ |
$32.48867972$ |
2.359324574 |
\( \frac{1217792}{135} a + \frac{4608704}{135} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -a - 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-a-2\right){x}$ |
135.2-b4 |
135.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 2^{12} \cdot 3^{11} \cdot 5^{3} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.377865667$ |
$1.804926651$ |
2.359324574 |
\( \frac{74932775168}{225} a + \frac{47391625792}{45} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -304 a - 970\) , \( -5610 a - 17760\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-304a-970\right){x}-5610a-17760$ |
135.2-c1 |
135.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 3^{12} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.193173311$ |
$13.37120092$ |
1.633606813 |
\( -\frac{6184497677}{3645} a + \frac{19558082533}{3645} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 4 a + 4\) , \( 5 a + 41\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(4a+4\right){x}+5a+41$ |
135.2-c2 |
135.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.386346623$ |
$26.74240184$ |
1.633606813 |
\( \frac{211922}{135} a + \frac{64105}{27} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-a-1\right){x}$ |
135.2-d1 |
135.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{18} \cdot 5^{4} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.198884253$ |
$3.102293440$ |
4.704572026 |
\( \frac{24897088}{18225} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -48 a + 174\) , \( 226 a - 676\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-48a+174\right){x}+226a-676$ |
135.2-d2 |
135.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{12} \cdot 5^{8} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.397768506$ |
$6.204586881$ |
4.704572026 |
\( \frac{36594368}{16875} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 15 a - 43\) , \( 21 a - 67\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(15a-43\right){x}+21a-67$ |
135.2-e1 |
135.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 3^{18} \cdot 5^{4} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.088005746$ |
$3.102293440$ |
2.072073465 |
\( \frac{24897088}{18225} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -15 a + 50\) , \( -7 a + 24\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a+50\right){x}-7a+24$ |
135.2-e2 |
135.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{12} \cdot 5^{8} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.044002873$ |
$6.204586881$ |
2.072073465 |
\( \frac{36594368}{16875} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 56 a - 190\) , \( 242 a - 728\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(56a-190\right){x}+242a-728$ |
135.2-f1 |
135.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 2^{12} \cdot 3^{12} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.174021352$ |
$13.37120092$ |
4.414933890 |
\( -\frac{6184497677}{3645} a + \frac{19558082533}{3645} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 13 a + 5\) , \( 48 a + 178\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13a+5\right){x}+48a+178$ |
135.2-f2 |
135.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 2^{12} \cdot 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.348042704$ |
$26.74240184$ |
4.414933890 |
\( \frac{211922}{135} a + \frac{64105}{27} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -7 a - 15\) , \( 8 a + 30\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-15\right){x}+8a+30$ |
135.2-g1 |
135.2-g |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.661192439$ |
$16.24433986$ |
2.844466074 |
\( \frac{1026304}{3645} a + \frac{949568}{729} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -4 a - 1\) , \( -3 a + 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-1\right){x}-3a+1$ |
135.2-g2 |
135.2-g |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{10} \cdot 5^{6} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$2.491788659$ |
$3.609853302$ |
2.844466074 |
\( -\frac{66006784}{375} a + \frac{226222784}{375} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -68 a - 266\) , \( -506 a - 1784\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-68a-266\right){x}-506a-1784$ |
135.2-g3 |
135.2-g |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{6} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.830596219$ |
$32.48867972$ |
2.844466074 |
\( \frac{1217792}{135} a + \frac{4608704}{135} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -8 a - 26\) , \( 38 a + 120\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8a-26\right){x}+38a+120$ |
135.2-g4 |
135.2-g |
$4$ |
$6$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( - 3^{11} \cdot 5^{3} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$4.983577319$ |
$1.804926651$ |
2.844466074 |
\( \frac{74932775168}{225} a + \frac{47391625792}{45} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -79 a - 241\) , \( -658 a - 2080\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-79a-241\right){x}-658a-2080$ |
135.2-h1 |
135.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{6} \cdot 5^{2} \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.140502805$ |
0.812784859 |
\( \frac{336226}{135} a - \frac{1062059}{135} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 7 a + 22\) , \( -154 a - 487\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(7a+22\right){x}-154a-487$ |
135.2-h2 |
135.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
135.2 |
\( 3^{3} \cdot 5 \) |
\( 2^{12} \cdot 3^{9} \cdot 5 \) |
$1.92642$ |
$(3,a+1), (3,a+2), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.140502805$ |
0.812784859 |
\( -\frac{156415766633}{3645} a + \frac{98945821613}{729} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -213 a - 678\) , \( -3366 a - 10647\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-213a-678\right){x}-3366a-10647$ |
*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.