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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 6080.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6080.j1 | 6080k3 | \([0, 0, 0, -8108, -281008]\) | \(899466517764/95\) | \(6225920\) | \([2]\) | \(4096\) | \(0.73189\) | |
6080.j2 | 6080k4 | \([0, 0, 0, -908, 3472]\) | \(1263284964/651605\) | \(42703585280\) | \([2]\) | \(4096\) | \(0.73189\) | |
6080.j3 | 6080k2 | \([0, 0, 0, -508, -4368]\) | \(884901456/9025\) | \(147865600\) | \([2, 2]\) | \(2048\) | \(0.38532\) | |
6080.j4 | 6080k1 | \([0, 0, 0, -8, -168]\) | \(-55296/11875\) | \(-12160000\) | \([2]\) | \(1024\) | \(0.038746\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6080.j have rank \(0\).
Complex multiplication
The elliptic curves in class 6080.j do not have complex multiplication.Modular form 6080.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.