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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1089e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
1089.g2 | 1089e1 | \([0, 0, 1, -66, -212]\) | \(-32768\) | \(-970299\) | \([]\) | \(120\) | \(-0.073767\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
1089.g1 | 1089e2 | \([0, 0, 1, -7986, 281839]\) | \(-32768\) | \(-1718943866739\) | \([]\) | \(1320\) | \(1.1252\) | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 1089e have rank \(0\).
Complex multiplication
Each elliptic curve in class 1089e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 1089.2.a.e
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.