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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 20825o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20825.u3 | 20825o1 | \([1, -1, 0, -842, -3809]\) | \(35937/17\) | \(31250515625\) | \([2]\) | \(9216\) | \(0.70789\) | \(\Gamma_0(N)\)-optimal |
| 20825.u2 | 20825o2 | \([1, -1, 0, -6967, 222816]\) | \(20346417/289\) | \(531258765625\) | \([2, 2]\) | \(18432\) | \(1.0545\) | |
| 20825.u1 | 20825o3 | \([1, -1, 0, -111092, 14279691]\) | \(82483294977/17\) | \(31250515625\) | \([2]\) | \(36864\) | \(1.4010\) | |
| 20825.u4 | 20825o4 | \([1, -1, 0, -842, 596441]\) | \(-35937/83521\) | \(-153533783265625\) | \([2]\) | \(36864\) | \(1.4010\) |
Rank
sage: E.rank()
The elliptic curves in class 20825o have rank \(1\).
Complex multiplication
The elliptic curves in class 20825o do not have complex multiplication.Modular form 20825.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
