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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1764.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1764.i1 | 1764g2 | \([0, 0, 0, -399, -2842]\) | \(109744/9\) | \(576108288\) | \([2]\) | \(768\) | \(0.42353\) | |
1764.i2 | 1764g1 | \([0, 0, 0, -84, 245]\) | \(16384/3\) | \(12002256\) | \([2]\) | \(384\) | \(0.076952\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1764.i have rank \(0\).
Complex multiplication
The elliptic curves in class 1764.i do not have complex multiplication.Modular form 1764.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.