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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 43681.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
43681.h1 | 43681c2 | \([0, 1, 1, -320327, 71490832]\) | \(-32768\) | \(-110931726475010771\) | \([]\) | \(316800\) | \(2.0481\) | \(-11\) | |
43681.h2 | 43681c1 | \([0, 1, 1, -2647, -54675]\) | \(-32768\) | \(-62618067611\) | \([]\) | \(28800\) | \(0.84915\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 43681.h have rank \(2\).
Complex multiplication
Each elliptic curve in class 43681.h has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 43681.2.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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.