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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 142912.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142912.f1 | 142912bh2 | \([0, 1, 0, -13153, -582561]\) | \(7680354317000/37530031\) | \(1229784055808\) | \([2]\) | \(196608\) | \(1.1679\) | |
142912.f2 | 142912bh1 | \([0, 1, 0, -393, -18569]\) | \(-1643032000/34904023\) | \(-142966878208\) | \([2]\) | \(98304\) | \(0.82134\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 142912.f have rank \(1\).
Complex multiplication
The elliptic curves in class 142912.f do not have complex multiplication.Modular form 142912.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.