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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 185150.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
185150.bc1 | 185150co4 | \([1, 1, 0, -2206340250, 39888289690250]\) | \(513516182162686336369/1944885031250\) | \(4498637259435727050781250\) | \([2]\) | \(189775872\) | \(3.9451\) | |
185150.bc2 | 185150co3 | \([1, 1, 0, -139934000, 603840471500]\) | \(131010595463836369/7704101562500\) | \(17820055058609008789062500\) | \([2]\) | \(94887936\) | \(3.5985\) | |
185150.bc3 | 185150co2 | \([1, 1, 0, -37572500, 9478025000]\) | \(2535986675931409/1450751712200\) | \(3355676866153111653125000\) | \([2]\) | \(63258624\) | \(3.3958\) | |
185150.bc4 | 185150co1 | \([1, 1, 0, -24347500, -46053750000]\) | \(690080604747409/3406760000\) | \(7880042893900625000000\) | \([2]\) | \(31629312\) | \(3.0492\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 185150.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 185150.bc do not have complex multiplication.Modular form 185150.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.