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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 72450.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.q1 | 72450bd2 | \([1, -1, 0, -213417, 37842741]\) | \(94376601570889/456435000\) | \(5199079921875000\) | \([2]\) | \(589824\) | \(1.8641\) | |
72450.q2 | 72450bd1 | \([1, -1, 0, -6417, 1203741]\) | \(-2565726409/53323200\) | \(-607384575000000\) | \([2]\) | \(294912\) | \(1.5176\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.q have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.q do not have complex multiplication.Modular form 72450.2.a.q
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.