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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8050c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8050.j4 | 8050c1 | \([1, 1, 0, -46025, 3765125]\) | \(690080604747409/3406760000\) | \(53230625000000\) | \([2]\) | \(59904\) | \(1.4815\) | \(\Gamma_0(N)\)-optimal |
8050.j3 | 8050c2 | \([1, 1, 0, -71025, -809875]\) | \(2535986675931409/1450751712200\) | \(22667995503125000\) | \([2]\) | \(119808\) | \(1.8280\) | |
8050.j2 | 8050c3 | \([1, 1, 0, -264525, -49744375]\) | \(131010595463836369/7704101562500\) | \(120376586914062500\) | \([2]\) | \(179712\) | \(2.0308\) | |
8050.j1 | 8050c4 | \([1, 1, 0, -4170775, -3280213125]\) | \(513516182162686336369/1944885031250\) | \(30388828613281250\) | \([2]\) | \(359424\) | \(2.3773\) |
Rank
sage: E.rank()
The elliptic curves in class 8050c have rank \(1\).
Complex multiplication
The elliptic curves in class 8050c do not have complex multiplication.Modular form 8050.2.a.c
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.