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Properties

Label 234135.z
Number of curves $2$
Conductor $234135$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("z1")
 
E.isogeny_class()
 

Elliptic curves in class 234135.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
234135.z1 234135z2 \([1, -1, 0, -151250325, 716004126250]\) \(8000051600110940079507/144453125\) \(6909503109609375\) \([2]\) \(22937600\) \(3.0302\)  
234135.z2 234135z1 \([1, -1, 0, -9453450, 11188579375]\) \(1953326569433829507/262451171875\) \(12553603033447265625\) \([2]\) \(11468800\) \(2.6836\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 234135.z have rank \(1\).

Complex multiplication

The elliptic curves in class 234135.z do not have complex multiplication.

Modular form 234135.2.a.z

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 2 q^{13} - 4 q^{14} - q^{16} + 6 q^{17} + 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.