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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 53130n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53130.l4 | 53130n1 | \([1, 1, 0, 2513, -38171]\) | \(1754006183281799/1679486054400\) | \(-1679486054400\) | \([2]\) | \(110592\) | \(1.0329\) | \(\Gamma_0(N)\)-optimal |
53130.l3 | 53130n2 | \([1, 1, 0, -13167, -361179]\) | \(252485100339244921/91458619560000\) | \(91458619560000\) | \([2, 2]\) | \(221184\) | \(1.3795\) | |
53130.l2 | 53130n3 | \([1, 1, 0, -90167, 10126221]\) | \(81072599558658172921/2290272305567400\) | \(2290272305567400\) | \([2]\) | \(442368\) | \(1.7260\) | |
53130.l1 | 53130n4 | \([1, 1, 0, -187047, -31207491]\) | \(723735009058769762041/198888834375000\) | \(198888834375000\) | \([2]\) | \(442368\) | \(1.7260\) |
Rank
sage: E.rank()
The elliptic curves in class 53130n have rank \(2\).
Complex multiplication
The elliptic curves in class 53130n do not have complex multiplication.Modular form 53130.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.