Properties

Label 75726d
Number of curves $2$
Conductor $75726$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 75726d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75726.g2 75726d1 \([1, -1, 0, -803619, -326622699]\) \(-78731237277328508209/17734929828102144\) \(-12928763844686462976\) \([]\) \(1749888\) \(2.3873\) \(\Gamma_0(N)\)-optimal
75726.g1 75726d2 \([1, -1, 0, -34230159, 99790755921]\) \(-6084472608417988103640049/2379033678299675392884\) \(-1734315551480463361412436\) \([]\) \(12249216\) \(3.3602\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75726d have rank \(1\).

Complex multiplication

The elliptic curves in class 75726d do not have complex multiplication.

Modular form 75726.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2 q^{11} - q^{14} + q^{16} - 4 q^{17} + 6 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.