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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 440062.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
440062.h1 | 440062h3 | \([1, -1, 0, -977543, 372091171]\) | \(16342588257633/8185058\) | \(51740723195121842\) | \([2]\) | \(4816896\) | \(2.1597\) | \(\Gamma_0(N)\)-optimal* |
440062.h2 | 440062h2 | \([1, -1, 0, -71533, 3707505]\) | \(6403769793/2775556\) | \(17545297138830244\) | \([2, 2]\) | \(2408448\) | \(1.8132\) | \(\Gamma_0(N)\)-optimal* |
440062.h3 | 440062h1 | \([1, -1, 0, -34553, -2423779]\) | \(721734273/13328\) | \(84251126717072\) | \([2]\) | \(1204224\) | \(1.4666\) | \(\Gamma_0(N)\)-optimal* |
440062.h4 | 440062h4 | \([1, -1, 0, 242797, 27282255]\) | \(250404380127/196003234\) | \(-1239007600892100466\) | \([2]\) | \(4816896\) | \(2.1597\) |
Rank
sage: E.rank()
The elliptic curves in class 440062.h have rank \(1\).
Complex multiplication
The elliptic curves in class 440062.h do not have complex multiplication.Modular form 440062.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.