Properties

Label 17238n
Number of curves $4$
Conductor $17238$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{12} - 2 q^{14} + q^{16} - q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.