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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 121605k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121605.e2 | 121605k1 | \([0, 1, 1, 28879, -277414]\) | \(1503484706816/890163675\) | \(-1576979250246675\) | \([]\) | \(518400\) | \(1.6055\) | \(\Gamma_0(N)\)-optimal |
121605.e1 | 121605k2 | \([0, 1, 1, -363161, 92920295]\) | \(-2989967081734144/380653171875\) | \(-674350313820046875\) | \([]\) | \(1555200\) | \(2.1548\) |
Rank
sage: E.rank()
The elliptic curves in class 121605k have rank \(2\).
Complex multiplication
The elliptic curves in class 121605k do not have complex multiplication.Modular form 121605.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.