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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 83205h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83205.q2 | 83205h1 | \([1, -1, 0, -144458094, 668242332783]\) | \(1953326569433829507/262451171875\) | \(44794326781549072265625\) | \([2]\) | \(16558080\) | \(3.3653\) | \(\Gamma_0(N)\)-optimal |
83205.q1 | 83205h2 | \([1, -1, 0, -2311254969, 42768672254658]\) | \(8000051600110940079507/144453125\) | \(24654797460564609375\) | \([2]\) | \(33116160\) | \(3.7118\) |
Rank
sage: E.rank()
The elliptic curves in class 83205h have rank \(1\).
Complex multiplication
The elliptic curves in class 83205h do not have complex multiplication.Modular form 83205.2.a.h
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.