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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4275.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4275.i1 | 4275k3 | \([0, 0, 1, -173100, 27720031]\) | \(-50357871050752/19\) | \(-216421875\) | \([]\) | \(7776\) | \(1.3875\) | |
4275.i2 | 4275k2 | \([0, 0, 1, -2100, 39406]\) | \(-89915392/6859\) | \(-78128296875\) | \([]\) | \(2592\) | \(0.83816\) | |
4275.i3 | 4275k1 | \([0, 0, 1, 150, 31]\) | \(32768/19\) | \(-216421875\) | \([]\) | \(864\) | \(0.28885\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4275.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4275.i do not have complex multiplication.Modular form 4275.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.