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SageMath
E = EllipticCurve("kd1")
E.isogeny_class()
Elliptic curves in class 284592.kd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
284592.kd1 | 284592kd2 | \([0, 1, 0, -264469, 52259843]\) | \(35084566528/1029\) | \(59999679860736\) | \([]\) | \(1990656\) | \(1.7436\) | |
284592.kd2 | 284592kd1 | \([0, 1, 0, -5749, -53341]\) | \(360448/189\) | \(11020349362176\) | \([]\) | \(663552\) | \(1.1943\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 284592.kd have rank \(1\).
Complex multiplication
The elliptic curves in class 284592.kd do not have complex multiplication.Modular form 284592.2.a.kd
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.