Properties

Label 5712.f
Number of curves $4$
Conductor $5712$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 5712.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.f1 5712g3 \([0, -1, 0, -181367424, -940067661600]\) \(322159999717985454060440834/4250799\) \(8705636352\) \([2]\) \(276480\) \(2.8910\)  
5712.f2 5712g4 \([0, -1, 0, -11364624, -14606337312]\) \(79260902459030376659234/842751810121431609\) \(1725955707128691935232\) \([4]\) \(276480\) \(2.8910\)  
5712.f3 5712g2 \([0, -1, 0, -11335464, -14685722496]\) \(157304700372188331121828/18069292138401\) \(18502955149722624\) \([2, 2]\) \(138240\) \(2.5444\)  
5712.f4 5712g1 \([0, -1, 0, -706644, -230527296]\) \(-152435594466395827792/1646846627220711\) \(-421592736568502016\) \([2]\) \(69120\) \(2.1979\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 5712.f have rank \(1\).

Complex multiplication

The elliptic curves in class 5712.f do not have complex multiplication.

Modular form 5712.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{7} + q^{9} + 2 q^{13} + 2 q^{15} + q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.