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SageMath
E = EllipticCurve("ht1")
E.isogeny_class()
Elliptic curves in class 414050.ht
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.ht1 | 414050ht1 | \([1, -1, 1, -556380, -159598253]\) | \(-5154200289/20\) | \(-73910512812500\) | \([]\) | \(6773760\) | \(1.8744\) | \(\Gamma_0(N)\)-optimal* |
414050.ht2 | 414050ht2 | \([1, -1, 1, 3879870, 1514642497]\) | \(1747829720511/1280000000\) | \(-4730272820000000000000\) | \([]\) | \(47416320\) | \(2.8473\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050.ht have rank \(1\).
Complex multiplication
The elliptic curves in class 414050.ht do not have complex multiplication.Modular form 414050.2.a.ht
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.