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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1575.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1575.j1 | 1575c2 | \([1, -1, 0, -582, -3799]\) | \(8869743/2401\) | \(5907360375\) | \([2]\) | \(768\) | \(0.58294\) | |
1575.j2 | 1575c1 | \([1, -1, 0, 93, -424]\) | \(35937/49\) | \(-120558375\) | \([2]\) | \(384\) | \(0.23637\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1575.j have rank \(1\).
Complex multiplication
The elliptic curves in class 1575.j do not have complex multiplication.Modular form 1575.2.a.j
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.