Properties

Label 8568c
Number of curves $4$
Conductor $8568$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 8568c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8568.j4 8568c1 \([0, 0, 0, -6359799, -6230596790]\) \(-152435594466395827792/1646846627220711\) \(-307341104958437969664\) \([2]\) \(276480\) \(2.7472\) \(\Gamma_0(N)\)-optimal
8568.j3 8568c2 \([0, 0, 0, -102019179, -396616526570]\) \(157304700372188331121828/18069292138401\) \(13488654304147792896\) \([2, 2]\) \(552960\) \(3.0937\)  
8568.j1 8568c3 \([0, 0, 0, -1632306819, -25383459170018]\) \(322159999717985454060440834/4250799\) \(6346408900608\) \([2]\) \(1105920\) \(3.4403\)  
8568.j2 8568c4 \([0, 0, 0, -102281619, -394473389042]\) \(79260902459030376659234/842751810121431609\) \(1258221710496816420784128\) \([2]\) \(1105920\) \(3.4403\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8568c have rank \(0\).

Complex multiplication

The elliptic curves in class 8568c do not have complex multiplication.

Modular form 8568.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.