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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 83790.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83790.o1 | 83790c4 | \([1, -1, 0, -52427265, -57470785219]\) | \(6882017790203934867/3366201047283200\) | \(7795062170953676616614400\) | \([2]\) | \(17915904\) | \(3.4696\) | |
83790.o2 | 83790c2 | \([1, -1, 0, -43000890, -108522798644]\) | \(2768241956450868452043/2058557375000\) | \(6539054848507125000\) | \([2]\) | \(5971968\) | \(2.9203\) | |
83790.o3 | 83790c1 | \([1, -1, 0, -2669970, -1718456300]\) | \(-662660286993086283/18441985352000\) | \(-58581390636291096000\) | \([2]\) | \(2985984\) | \(2.5737\) | \(\Gamma_0(N)\)-optimal |
83790.o4 | 83790c3 | \([1, -1, 0, 11923455, -6878249155]\) | \(80956273702840173/55667967918080\) | \(-128909493151726518927360\) | \([2]\) | \(8957952\) | \(3.1231\) |
Rank
sage: E.rank()
The elliptic curves in class 83790.o have rank \(0\).
Complex multiplication
The elliptic curves in class 83790.o do not have complex multiplication.Modular form 83790.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.