Properties

Label 234234.em
Number of curves $4$
Conductor $234234$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("em1")
 
E.isogeny_class()
 

Elliptic curves in class 234234.em

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
234234.em1 234234em4 \([1, -1, 1, -21496325, 38364079511]\) \(312196988566716625/25367712678\) \(89262480716553101958\) \([2]\) \(10616832\) \(2.8733\)  
234234.em2 234234em3 \([1, -1, 1, -1251815, 684997499]\) \(-61653281712625/21875235228\) \(-76973347478932412508\) \([2]\) \(5308416\) \(2.5268\)  
234234.em3 234234em2 \([1, -1, 1, -552155, -79067725]\) \(5290763640625/2291573592\) \(8063460279722359512\) \([2]\) \(3538944\) \(2.3240\)  
234234.em4 234234em1 \([1, -1, 1, 117085, -9199069]\) \(50447927375/39517632\) \(-139052421049493952\) \([2]\) \(1769472\) \(1.9775\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 234234.em have rank \(1\).

Complex multiplication

The elliptic curves in class 234234.em do not have complex multiplication.

Modular form 234234.2.a.em

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{7} + q^{8} - q^{11} - q^{14} + q^{16} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.