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SageMath
E = EllipticCurve("ib1")
E.isogeny_class()
Elliptic curves in class 476850.ib
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.ib1 | 476850ib8 | \([1, 0, 0, -1453933586213, -674784417438323583]\) | \(901247067798311192691198986281/552431869440\) | \(208349411975109240000000\) | \([2]\) | \(4586471424\) | \(5.1821\) | |
476850.ib2 | 476850ib7 | \([1, 0, 0, -91481234213, -10394687505395583]\) | \(224494757451893010998773801/6152490825146276160000\) | \(2320408934591174623516485000000000\) | \([2]\) | \(4586471424\) | \(5.1821\) | |
476850.ib3 | 476850ib6 | \([1, 0, 0, -90870866213, -10543508041523583]\) | \(220031146443748723000125481/172266701724057600\) | \(64970303113544676249600000000\) | \([2, 2]\) | \(2293235712\) | \(4.8355\) | |
476850.ib4 | 476850ib5 | \([1, 0, 0, -17953456838, -925235231226708]\) | \(1696892787277117093383481/1440538624914939000\) | \(543298444625772800676421875000\) | \([2]\) | \(1528823808\) | \(4.6328\) | |
476850.ib5 | 476850ib4 | \([1, 0, 0, -11757874838, 485484947651292]\) | \(476646772170172569823801/5862293314453125000\) | \(2210961083997671905517578125000\) | \([2]\) | \(1528823808\) | \(4.6328\) | \(\Gamma_0(N)\)-optimal* |
476850.ib6 | 476850ib3 | \([1, 0, 0, -5641298213, -167063826227583]\) | \(-52643812360427830814761/1504091705903677440\) | \(-567267458337151899402240000000\) | \([4]\) | \(1146617856\) | \(4.4890\) | |
476850.ib7 | 476850ib2 | \([1, 0, 0, -1372081838, -7539031851708]\) | \(757443433548897303481/373234243041000000\) | \(140765113977576677015625000000\) | \([2, 2]\) | \(764411904\) | \(4.2862\) | \(\Gamma_0(N)\)-optimal* |
476850.ib8 | 476850ib1 | \([1, 0, 0, 313366162, -903423075708]\) | \(9023321954633914439/6156756739584000\) | \(-2322017822155059864000000000\) | \([4]\) | \(382205952\) | \(3.9397\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 476850.ib have rank \(1\).
Complex multiplication
The elliptic curves in class 476850.ib do not have complex multiplication.Modular form 476850.2.a.ib
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.