Properties

Label 8470.k
Number of curves $2$
Conductor $8470$
CM no
Rank $0$
Graph

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

Elliptic curves in class 8470.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.k1 8470c1 \([1, 0, 1, -51549, -4518384]\) \(-584043889/1400\) \(-36312394441400\) \([]\) \(38016\) \(1.4813\) \(\Gamma_0(N)\)-optimal
8470.k2 8470c2 \([1, 0, 1, 94861, -22731788]\) \(3639707951/10718750\) \(-278016769941968750\) \([]\) \(114048\) \(2.0306\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8470.k have rank \(0\).

Complex multiplication

The elliptic curves in class 8470.k do not have complex multiplication.

Modular form 8470.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} - 2 q^{9} + q^{10} + q^{12} - 5 q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} + 2 q^{18} - 5 q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.