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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 150480ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 150480.bo7 | 150480ce1 | \([0, 0, 0, -12457083, 14851322282]\) | \(71595431380957421881/9522562500000000\) | \(28434219264000000000000\) | \([2]\) | \(12386304\) | \(3.0362\) | \(\Gamma_0(N)\)-optimal |
| 150480.bo5 | 150480ce2 | \([0, 0, 0, -192457083, 1027639322282]\) | \(264020672568758737421881/5803468580250000\) | \(17329064325129216000000\) | \([2, 2]\) | \(24772608\) | \(3.3828\) | |
| 150480.bo4 | 150480ce3 | \([0, 0, 0, -251407083, -1532372607718]\) | \(588530213343917460371881/861551575695360000\) | \(2572579220201133834240000\) | \([2]\) | \(37158912\) | \(3.5856\) | |
| 150480.bo2 | 150480ce4 | \([0, 0, 0, -3079297083, 65769645266282]\) | \(1081411559614045490773061881/522522049500\) | \(1560242479454208000\) | \([2]\) | \(49545216\) | \(3.7294\) | |
| 150480.bo6 | 150480ce5 | \([0, 0, 0, -185617083, 1104065378282]\) | \(-236859095231405581781881/39282983014374049500\) | \(-117298358753192681822208000\) | \([2]\) | \(49545216\) | \(3.7294\) | |
| 150480.bo3 | 150480ce6 | \([0, 0, 0, -325135083, -560150962918]\) | \(1272998045160051207059881/691293848290254950400\) | \(2064192370293128637815193600\) | \([2, 2]\) | \(74317824\) | \(3.9321\) | |
| 150480.bo1 | 150480ce7 | \([0, 0, 0, -3085096683, 65509465794842]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(25333852109399382516368670720\) | \([2]\) | \(148635648\) | \(4.2787\) | |
| 150480.bo8 | 150480ce8 | \([0, 0, 0, 1255178517, -4407582453478]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-134952370369413401316545003520\) | \([2]\) | \(148635648\) | \(4.2787\) |
Rank
sage: E.rank()
The elliptic curves in class 150480ce have rank \(1\).
Complex multiplication
The elliptic curves in class 150480ce do not have complex multiplication.Modular form 150480.2.a.ce
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
