Properties

Label 188598.l
Number of curves $4$
Conductor $188598$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 188598.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
188598.l1 188598k3 \([1, 1, 1, -1387713, 475128639]\) \(46753267515625/11591221248\) \(73272317689890865152\) \([2]\) \(5806080\) \(2.5220\)  
188598.l2 188598k1 \([1, 1, 1, -472458, -125145513]\) \(1845026709625/793152\) \(5013801745040448\) \([2]\) \(1935360\) \(1.9727\) \(\Gamma_0(N)\)-optimal
188598.l3 188598k2 \([1, 1, 1, -398498, -165557257]\) \(-1107111813625/1228691592\) \(-7767005628285784008\) \([2]\) \(3870720\) \(2.3193\)  
188598.l4 188598k4 \([1, 1, 1, 3345727, 3017932607]\) \(655215969476375/1001033261568\) \(-6327894670495907000832\) \([2]\) \(11612160\) \(2.8686\)  

Rank

sage: E.rank()
 

The elliptic curves in class 188598.l have rank \(1\).

Complex multiplication

The elliptic curves in class 188598.l do not have complex multiplication.

Modular form 188598.2.a.l

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9} - q^{12} + 2 q^{13} - 2 q^{14} + q^{16} - q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.