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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 180708f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180708.d2 | 180708f1 | \([0, -1, 0, -105869, 15446934]\) | \(-3196715008/649539\) | \(-26664629855607216\) | \([2]\) | \(1520640\) | \(1.8742\) | \(\Gamma_0(N)\)-optimal |
| 180708.d1 | 180708f2 | \([0, -1, 0, -1769204, 906329160]\) | \(932410994128/29403\) | \(19312653722579712\) | \([2]\) | \(3041280\) | \(2.2208\) |
Rank
sage: E.rank()
The elliptic curves in class 180708f have rank \(0\).
Complex multiplication
The elliptic curves in class 180708f do not have complex multiplication.Modular form 180708.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
