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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 430950.hc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
430950.hc1 | 430950hc2 | \([1, 0, 0, -12497638, -17006566708]\) | \(-71559517896165625/4598568\) | \(-13872755880945000\) | \([]\) | \(14805504\) | \(2.5563\) | |
430950.hc2 | 430950hc1 | \([1, 0, 0, -139513, -27985933]\) | \(-99546915625/54454842\) | \(-164276950911986250\) | \([]\) | \(4935168\) | \(2.0070\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 430950.hc have rank \(0\).
Complex multiplication
The elliptic curves in class 430950.hc do not have complex multiplication.Modular form 430950.2.a.hc
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.