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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 17238n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
17238.p2 | 17238n1 | \([1, 0, 0, -43183, 3449081]\) | \(1845026709625/793152\) | \(3828393211968\) | \([2]\) | \(55296\) | \(1.3746\) | \(\Gamma_0(N)\)-optimal |
17238.p3 | 17238n2 | \([1, 0, 0, -36423, 4567185]\) | \(-1107111813625/1228691592\) | \(-5930659634489928\) | \([2]\) | \(110592\) | \(1.7212\) | |
17238.p1 | 17238n3 | \([1, 0, 0, -126838, -13155676]\) | \(46753267515625/11591221248\) | \(55948611040837632\) | \([2]\) | \(165888\) | \(1.9239\) | |
17238.p4 | 17238n4 | \([1, 0, 0, 305802, -83329884]\) | \(655215969476375/1001033261568\) | \(-4831796356235776512\) | \([2]\) | \(331776\) | \(2.2705\) |
Rank
sage: E.rank()
The elliptic curves in class 17238n have rank \(1\).
Complex multiplication
The elliptic curves in class 17238n do not have complex multiplication.Modular form 17238.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.