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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12138.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12138.f1 | 12138b1 | \([1, 1, 0, -31, 67]\) | \(-11984473/2646\) | \(-764694\) | \([]\) | \(2592\) | \(-0.15052\) | \(\Gamma_0(N)\)-optimal |
12138.f2 | 12138b2 | \([1, 1, 0, 224, -392]\) | \(4271073047/2823576\) | \(-816013464\) | \([]\) | \(7776\) | \(0.39879\) |
Rank
sage: E.rank()
The elliptic curves in class 12138.f have rank \(1\).
Complex multiplication
The elliptic curves in class 12138.f do not have complex multiplication.Modular form 12138.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.