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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 12075.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12075.k1 | 12075g1 | \([0, -1, 1, -533, 14843]\) | \(-1073741824/5325075\) | \(-83204296875\) | \([]\) | \(10368\) | \(0.77981\) | \(\Gamma_0(N)\)-optimal |
12075.k2 | 12075g2 | \([0, -1, 1, 4717, -360532]\) | \(742692847616/3992296875\) | \(-62379638671875\) | \([]\) | \(31104\) | \(1.3291\) |
Rank
sage: E.rank()
The elliptic curves in class 12075.k have rank \(0\).
Complex multiplication
The elliptic curves in class 12075.k do not have complex multiplication.Modular form 12075.2.a.k
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.