Properties

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

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

Elliptic curves in class 8470.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.d1 8470g2 \([1, 0, 1, -179, 306]\) \(472729139/240100\) \(319573100\) \([2]\) \(4608\) \(0.32433\)  
8470.d2 8470g1 \([1, 0, 1, 41, 42]\) \(5929741/3920\) \(-5217520\) \([2]\) \(2304\) \(-0.022244\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8470.d have rank \(2\).

Complex multiplication

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

Modular form 8470.2.a.d

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.