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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 94864.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
94864.bk1 | 94864bt2 | \([0, -1, 0, -695669, -228913859]\) | \(-32768\) | \(-1136272129632088064\) | \([]\) | \(1216512\) | \(2.2420\) | \(-11\) | |
94864.bk2 | 94864bt1 | \([0, -1, 0, -5749, 174077]\) | \(-32768\) | \(-641395994624\) | \([]\) | \(110592\) | \(1.0430\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 94864.bk have rank \(1\).
Complex multiplication
Each elliptic curve in class 94864.bk has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 94864.2.a.bk
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.