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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 72450k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.s2 | 72450k1 | \([1, -1, 0, -33117, 1951541]\) | \(104487111/18032\) | \(693210656250000\) | \([2]\) | \(337920\) | \(1.5679\) | \(\Gamma_0(N)\)-optimal |
72450.s1 | 72450k2 | \([1, -1, 0, -505617, 138504041]\) | \(371850068871/14812\) | \(569423039062500\) | \([2]\) | \(675840\) | \(1.9145\) |
Rank
sage: E.rank()
The elliptic curves in class 72450k have rank \(0\).
Complex multiplication
The elliptic curves in class 72450k do not have complex multiplication.Modular form 72450.2.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.