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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 8112.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8112.be1 | 8112k5 | \([0, 1, 0, -64952, 6349812]\) | \(3065617154/9\) | \(88967743488\) | \([2]\) | \(18432\) | \(1.3303\) | |
8112.be2 | 8112k3 | \([0, 1, 0, -10872, -439932]\) | \(28756228/3\) | \(14827957248\) | \([2]\) | \(9216\) | \(0.98370\) | |
8112.be3 | 8112k4 | \([0, 1, 0, -4112, 95460]\) | \(1556068/81\) | \(400354845696\) | \([2, 2]\) | \(9216\) | \(0.98370\) | |
8112.be4 | 8112k2 | \([0, 1, 0, -732, -5940]\) | \(35152/9\) | \(11120967936\) | \([2, 2]\) | \(4608\) | \(0.63712\) | |
8112.be5 | 8112k1 | \([0, 1, 0, 113, -532]\) | \(2048/3\) | \(-231686832\) | \([2]\) | \(2304\) | \(0.29055\) | \(\Gamma_0(N)\)-optimal |
8112.be6 | 8112k6 | \([0, 1, 0, 2648, 384788]\) | \(207646/6561\) | \(-64857485002752\) | \([2]\) | \(18432\) | \(1.3303\) |
Rank
sage: E.rank()
The elliptic curves in class 8112.be have rank \(0\).
Complex multiplication
The elliptic curves in class 8112.be do not have complex multiplication.Modular form 8112.2.a.be
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.