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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 123627.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123627.j1 | 123627h2 | \([1, 1, 1, -6244022, -1416871996]\) | \(1121622319/613089\) | \(14716138673653379338383\) | \([2]\) | \(6773760\) | \(2.9444\) | |
123627.j2 | 123627h1 | \([1, 1, 1, -4801707, -4046500704]\) | \(510082399/783\) | \(18794557693043907201\) | \([2]\) | \(3386880\) | \(2.5978\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123627.j have rank \(0\).
Complex multiplication
The elliptic curves in class 123627.j do not have complex multiplication.Modular form 123627.2.a.j
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.