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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 331200.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.r1 | 331200r2 | \([0, 0, 0, -770700, -237994000]\) | \(33909572018/3234375\) | \(4828896000000000000\) | \([2]\) | \(8257536\) | \(2.3229\) | |
331200.r2 | 331200r1 | \([0, 0, 0, 57300, -17746000]\) | \(27871484/198375\) | \(-148086144000000000\) | \([2]\) | \(4128768\) | \(1.9763\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 331200.r have rank \(1\).
Complex multiplication
The elliptic curves in class 331200.r do not have complex multiplication.Modular form 331200.2.a.r
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.