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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 53130e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53130.d4 | 53130e1 | \([1, 1, 0, 137, 277]\) | \(281140102151/183617280\) | \(-183617280\) | \([2]\) | \(22528\) | \(0.27455\) | \(\Gamma_0(N)\)-optimal |
53130.d3 | 53130e2 | \([1, 1, 0, -583, 1573]\) | \(21973174804729/11291187600\) | \(11291187600\) | \([2, 2]\) | \(45056\) | \(0.62113\) | |
53130.d2 | 53130e3 | \([1, 1, 0, -5203, -145343]\) | \(15581727508423609/161608177500\) | \(161608177500\) | \([2]\) | \(90112\) | \(0.96770\) | |
53130.d1 | 53130e4 | \([1, 1, 0, -7483, 245833]\) | \(46349134440566329/48511196580\) | \(48511196580\) | \([2]\) | \(90112\) | \(0.96770\) |
Rank
sage: E.rank()
The elliptic curves in class 53130e have rank \(2\).
Complex multiplication
The elliptic curves in class 53130e do not have complex multiplication.Modular form 53130.2.a.e
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.