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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 133200.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
133200.k1 | 133200bq1 | \([0, 0, 0, -1711875, 2060181250]\) | \(-19026212425/51868672\) | \(-1512490475520000000000\) | \([]\) | \(5184000\) | \(2.7514\) | \(\Gamma_0(N)\)-optimal |
133200.k2 | 133200bq2 | \([0, 0, 0, 14938125, -46774268750]\) | \(12642252501575/39728447488\) | \(-1158481528750080000000000\) | \([]\) | \(15552000\) | \(3.3007\) |
Rank
sage: E.rank()
The elliptic curves in class 133200.k have rank \(1\).
Complex multiplication
The elliptic curves in class 133200.k do not have complex multiplication.Modular form 133200.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.