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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 165649g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
165649.g2 | 165649g1 | \([0, -1, 1, -10039, 400597]\) | \(-32768\) | \(-3414981850379\) | \([]\) | \(200880\) | \(1.1824\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
165649.g1 | 165649g2 | \([0, -1, 1, -1214759, -528335952]\) | \(-32768\) | \(-6049848661839271619\) | \([]\) | \(2209680\) | \(2.3813\) | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 165649g have rank \(1\).
Complex multiplication
Each elliptic curve in class 165649g has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 165649.2.a.g
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.