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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4680k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4680.f2 | 4680k1 | \([0, 0, 0, -243, 4158]\) | \(-78732/325\) | \(-6550502400\) | \([2]\) | \(2304\) | \(0.56877\) | \(\Gamma_0(N)\)-optimal |
4680.f1 | 4680k2 | \([0, 0, 0, -5643, 162918]\) | \(492983766/845\) | \(34062612480\) | \([2]\) | \(4608\) | \(0.91535\) |
Rank
sage: E.rank()
The elliptic curves in class 4680k have rank \(1\).
Complex multiplication
The elliptic curves in class 4680k do not have complex multiplication.Modular form 4680.2.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 Cremona 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 Cremona labels.