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SageMath
E = EllipticCurve("gi1")
E.isogeny_class()
Elliptic curves in class 121275gi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121275.dy1 | 121275gi1 | \([0, 0, 1, -257250, 52682656]\) | \(-56197120/3267\) | \(-109452311448046875\) | \([]\) | \(1088640\) | \(2.0258\) | \(\Gamma_0(N)\)-optimal |
121275.dy2 | 121275gi2 | \([0, 0, 1, 1396500, 93199531]\) | \(8990228480/5314683\) | \(-178054587990094921875\) | \([]\) | \(3265920\) | \(2.5751\) |
Rank
sage: E.rank()
The elliptic curves in class 121275gi have rank \(1\).
Complex multiplication
The elliptic curves in class 121275gi do not have complex multiplication.Modular form 121275.2.a.gi
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.