Properties

Label 8470.z
Number of curves $4$
Conductor $8470$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.z1 8470ba4 \([1, -1, 1, -2530012, 1549562831]\) \(1010962818911303721/57392720\) \(101674704435920\) \([2]\) \(122880\) \(2.1543\)  
8470.z2 8470ba3 \([1, -1, 1, -264892, -12309041]\) \(1160306142246441/634128110000\) \(1123396628679710000\) \([2]\) \(122880\) \(2.1543\)  
8470.z3 8470ba2 \([1, -1, 1, -158412, 24149711]\) \(248158561089321/1859334400\) \(3293924308998400\) \([2, 2]\) \(61440\) \(1.8078\)  
8470.z4 8470ba1 \([1, -1, 1, -3532, 855759]\) \(-2749884201/176619520\) \(-312892253470720\) \([4]\) \(30720\) \(1.4612\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8470.2.a.z

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.