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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 13005.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13005.j1 | 13005m2 | \([0, 0, 1, -7752, 262705]\) | \(244534214656/5\) | \(1053405\) | \([]\) | \(7776\) | \(0.68615\) | |
13005.j2 | 13005m1 | \([0, 0, 1, -102, 310]\) | \(557056/125\) | \(26335125\) | \([]\) | \(2592\) | \(0.13684\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13005.j have rank \(1\).
Complex multiplication
The elliptic curves in class 13005.j do not have complex multiplication.Modular form 13005.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.