Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.c1 8470f4 \([1, 0, 1, -3160039, 2100623886]\) \(1969902499564819009/63690429687500\) \(112831481307617187500\) \([2]\) \(414720\) \(2.6216\)  
8470.c2 8470f2 \([1, 0, 1, -432699, -108616378]\) \(5057359576472449/51765560000\) \(91705847239160000\) \([2]\) \(138240\) \(2.0723\)  
8470.c3 8470f1 \([1, 0, 1, -6779, -4180794]\) \(-19443408769/4249907200\) \(-7528969849139200\) \([2]\) \(69120\) \(1.7257\) \(\Gamma_0(N)\)-optimal
8470.c4 8470f3 \([1, 0, 1, 60981, 112610342]\) \(14156681599871/3100231750000\) \(-5492249659261750000\) \([2]\) \(207360\) \(2.2750\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8470.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.