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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 494190co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
494190.co4 | 494190co1 | \([1, -1, 0, 24280281, 97911713533]\) | \(89962967236397039/287450726400000\) | \(-5058065710340908646400000\) | \([2]\) | \(98304000\) | \(3.4235\) | \(\Gamma_0(N)\)-optimal* |
494190.co3 | 494190co2 | \([1, -1, 0, -228744999, 1147308759805]\) | \(75224183150104868881/11219310000000000\) | \(197418207688637310000000000\) | \([2]\) | \(196608000\) | \(3.7700\) | \(\Gamma_0(N)\)-optimal* |
494190.co2 | 494190co3 | \([1, -1, 0, -8587110519, 306282282600973]\) | \(-3979640234041473454886161/1471455901872240\) | \(-25892161535823949772544240\) | \([2]\) | \(491520000\) | \(4.2282\) | \(\Gamma_0(N)\)-optimal* |
494190.co1 | 494190co4 | \([1, -1, 0, -137393780499, 19601959388282905]\) | \(16300610738133468173382620881/2228489100\) | \(39213135564991469100\) | \([2]\) | \(983040000\) | \(4.5747\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 494190co have rank \(1\).
Complex multiplication
The elliptic curves in class 494190co do not have complex multiplication.Modular form 494190.2.a.co
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.