Properties

Label 8712k
Number of curves $4$
Conductor $8712$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8712k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8712.f3 8712k1 \([0, 0, 0, -13431, -577654]\) \(810448/33\) \(10910321402112\) \([2]\) \(15360\) \(1.2675\) \(\Gamma_0(N)\)-optimal
8712.f2 8712k2 \([0, 0, 0, -35211, 1770230]\) \(3650692/1089\) \(1440162425078784\) \([2, 2]\) \(30720\) \(1.6141\)  
8712.f1 8712k3 \([0, 0, 0, -514371, 141972446]\) \(5690357426/891\) \(2356629422856192\) \([2]\) \(61440\) \(1.9607\)  
8712.f4 8712k4 \([0, 0, 0, 95469, 11832590]\) \(36382894/43923\) \(-116173102289688576\) \([2]\) \(61440\) \(1.9607\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8712k have rank \(1\).

Complex multiplication

The elliptic curves in class 8712k do not have complex multiplication.

Modular form 8712.2.a.k

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - 2q^{13} + 6q^{17} + O(q^{20})\)  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.