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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 15680.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15680.k1 | 15680x2 | \([0, 1, 0, -961, 7839]\) | \(2185454/625\) | \(28098560000\) | \([2]\) | \(12288\) | \(0.71147\) | |
15680.k2 | 15680x1 | \([0, 1, 0, 159, 895]\) | \(19652/25\) | \(-561971200\) | \([2]\) | \(6144\) | \(0.36490\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15680.k have rank \(2\).
Complex multiplication
The elliptic curves in class 15680.k do not have complex multiplication.Modular form 15680.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.