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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 32487m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32487.e4 | 32487m1 | \([1, 0, 0, -80557, -8807128]\) | \(491411892194497/78897\) | \(9282153153\) | \([2]\) | \(73728\) | \(1.3151\) | \(\Gamma_0(N)\)-optimal |
| 32487.e3 | 32487m2 | \([1, 0, 0, -80802, -8750925]\) | \(495909170514577/6224736609\) | \(732334037312241\) | \([2, 2]\) | \(147456\) | \(1.6617\) | |
| 32487.e5 | 32487m3 | \([1, 0, 0, -13917, -22783398]\) | \(-2533811507137/1904381781393\) | \(-224048612199105057\) | \([2]\) | \(294912\) | \(2.0083\) | |
| 32487.e2 | 32487m4 | \([1, 0, 0, -151607, 8879520]\) | \(3275619238041697/1605271262049\) | \(188858558708802801\) | \([2, 2]\) | \(294912\) | \(2.0083\) | |
| 32487.e6 | 32487m5 | \([1, 0, 0, 552278, 68146637]\) | \(158346567380527343/108665074944153\) | \(-12784337402104656297\) | \([2]\) | \(589824\) | \(2.3548\) | |
| 32487.e1 | 32487m6 | \([1, 0, 0, -1988372, 1078244103]\) | \(7389727131216686257/6115533215337\) | \(719486367251182713\) | \([2]\) | \(589824\) | \(2.3548\) |
Rank
sage: E.rank()
The elliptic curves in class 32487m have rank \(1\).
Complex multiplication
The elliptic curves in class 32487m do not have complex multiplication.Modular form 32487.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
