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SageMath
E = EllipticCurve("ey1")
E.isogeny_class()
Elliptic curves in class 364650.ey
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
364650.ey1 | 364650ey3 | \([1, 1, 1, -82380513, -287781855969]\) | \(3957101249824708884951625/772310238681366528\) | \(12067347479396352000000\) | \([2]\) | \(65691648\) | \(3.2368\) | |
364650.ey2 | 364650ey4 | \([1, 1, 1, -73676513, -350955487969]\) | \(-2830680648734534916567625/1766676274677722124288\) | \(-27604316791839408192000000\) | \([2]\) | \(131383296\) | \(3.5834\) | |
364650.ey3 | 364650ey1 | \([1, 1, 1, -2508138, 988967031]\) | \(111675519439697265625/37528570137307392\) | \(586383908395428000000\) | \([2]\) | \(21897216\) | \(2.6875\) | \(\Gamma_0(N)\)-optimal |
364650.ey4 | 364650ey2 | \([1, 1, 1, 7317862, 6845263031]\) | \(2773679829880629422375/2899504554614368272\) | \(-45304758665849504250000\) | \([2]\) | \(43794432\) | \(3.0341\) |
Rank
sage: E.rank()
The elliptic curves in class 364650.ey have rank \(1\).
Complex multiplication
The elliptic curves in class 364650.ey do not have complex multiplication.Modular form 364650.2.a.ey
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.