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Properties

Label 151725z
Number of curves $2$
Conductor $151725$
CM no
Rank $1$
Graph

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

Elliptic curves in class 151725z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
151725.x2 151725z1 \([1, 1, 1, -1638, 12906]\) \(50653/21\) \(201509765625\) \([2]\) \(204800\) \(0.86635\) \(\Gamma_0(N)\)-optimal
151725.x1 151725z2 \([1, 1, 1, -12263, -518344]\) \(21253933/441\) \(4231705078125\) \([2]\) \(409600\) \(1.2129\)  

Rank

sage: E.rank()
 

The elliptic curves in class 151725z have rank \(1\).

Complex multiplication

The elliptic curves in class 151725z do not have complex multiplication.

Modular form 151725.2.a.z

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + q^{7} + 3 q^{8} + q^{9} + 4 q^{11} + q^{12} + 6 q^{13} - q^{14} - q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.