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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 407330u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
407330.u3 | 407330u1 | \([1, 0, 0, -29635, -38214975]\) | \(-19443408769/4249907200\) | \(-629138790519500800\) | \([2]\) | \(7299072\) | \(2.0945\) | \(\Gamma_0(N)\)-optimal* |
407330.u2 | 407330u2 | \([1, 0, 0, -1891715, -992717183]\) | \(5057359576472449/51765560000\) | \(7663160694182840000\) | \([2]\) | \(14598144\) | \(2.4411\) | \(\Gamma_0(N)\)-optimal* |
407330.u4 | 407330u3 | \([1, 0, 0, 266605, 1029374737]\) | \(14156681599871/3100231750000\) | \(-458945563217275750000\) | \([2]\) | \(21897216\) | \(2.6438\) | \(\Gamma_0(N)\)-optimal* |
407330.u1 | 407330u4 | \([1, 0, 0, -13815375, 19203578125]\) | \(1969902499564819009/63690429687500\) | \(9428469379581054687500\) | \([2]\) | \(43794432\) | \(2.9904\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 407330u have rank \(1\).
Complex multiplication
The elliptic curves in class 407330u do not have complex multiplication.Modular form 407330.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.