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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 42135.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42135.k1 | 42135f8 | \([1, 0, 1, -6067499, -5753085163]\) | \(1114544804970241/405\) | \(8976566257245\) | \([2]\) | \(585728\) | \(2.2760\) | |
42135.k2 | 42135f6 | \([1, 0, 1, -379274, -89888353]\) | \(272223782641/164025\) | \(3635509334184225\) | \([2, 2]\) | \(292864\) | \(1.9294\) | |
42135.k3 | 42135f7 | \([1, 0, 1, -309049, -124186243]\) | \(-147281603041/215233605\) | \(-4770515348316540045\) | \([2]\) | \(585728\) | \(2.2760\) | |
42135.k4 | 42135f4 | \([1, 0, 1, -224779, 40999811]\) | \(56667352321/15\) | \(332465416935\) | \([2]\) | \(146432\) | \(1.5829\) | |
42135.k5 | 42135f3 | \([1, 0, 1, -28149, -843053]\) | \(111284641/50625\) | \(1122070782155625\) | \([2, 2]\) | \(146432\) | \(1.5829\) | |
42135.k6 | 42135f2 | \([1, 0, 1, -14104, 634481]\) | \(13997521/225\) | \(4986981254025\) | \([2, 2]\) | \(73216\) | \(1.2363\) | |
42135.k7 | 42135f1 | \([1, 0, 1, -59, 27737]\) | \(-1/15\) | \(-332465416935\) | \([2]\) | \(36608\) | \(0.88972\) | \(\Gamma_0(N)\)-optimal |
42135.k8 | 42135f5 | \([1, 0, 1, 98256, -6303749]\) | \(4733169839/3515625\) | \(-77921582094140625\) | \([2]\) | \(292864\) | \(1.9294\) |
Rank
sage: E.rank()
The elliptic curves in class 42135.k have rank \(0\).
Complex multiplication
The elliptic curves in class 42135.k do not have complex multiplication.Modular form 42135.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.