Properties

Label 8470.r
Number of curves $4$
Conductor $8470$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.r1 8470u4 \([1, 0, 0, -1893471, -774028199]\) \(423783056881319689/99207416000000\) \(175751989096376000000\) \([2]\) \(414720\) \(2.5965\)  
8470.r2 8470u2 \([1, 0, 0, -1770656, -907026680]\) \(346553430870203929/8300600\) \(14705019236600\) \([2]\) \(138240\) \(2.0472\)  
8470.r3 8470u1 \([1, 0, 0, -110536, -14214144]\) \(-84309998289049/414124480\) \(-733646777913280\) \([2]\) \(69120\) \(1.7006\) \(\Gamma_0(N)\)-optimal
8470.r4 8470u3 \([1, 0, 0, 274849, -75395495]\) \(1296134247276791/2137096192000\) \(-3785996266995712000\) \([2]\) \(207360\) \(2.2499\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8470.r have rank \(1\).

Complex multiplication

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

Modular form 8470.2.a.r

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.