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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 34969a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
34969.j2 | 34969a1 | \([0, 1, 1, -2119, 37824]\) | \(-32768\) | \(-32127104339\) | \([]\) | \(20736\) | \(0.79353\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
34969.j1 | 34969a2 | \([0, 1, 1, -256439, -51369785]\) | \(-32768\) | \(-56915125089903179\) | \([]\) | \(228096\) | \(1.9925\) | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 34969a have rank \(1\).
Complex multiplication
Each elliptic curve in class 34969a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 34969.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.