Properties

Label 70.a
Number of curves $4$
Conductor $70$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 70.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
70.a1 70a3 \([1, -1, 1, -268, -1619]\) \(2121328796049/120050\) \(120050\) \([2]\) \(16\) \(0.038988\)  
70.a2 70a4 \([1, -1, 1, -88, 317]\) \(74565301329/5468750\) \(5468750\) \([2]\) \(16\) \(0.038988\)  
70.a3 70a2 \([1, -1, 1, -18, -19]\) \(611960049/122500\) \(122500\) \([2, 2]\) \(8\) \(-0.30759\)  
70.a4 70a1 \([1, -1, 1, 2, -3]\) \(1367631/2800\) \(-2800\) \([4]\) \(4\) \(-0.65416\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 70.a have rank \(0\).

Complex multiplication

The elliptic curves in class 70.a do not have complex multiplication.

Modular form 70.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3q^{9} - q^{10} + 4q^{11} - 6q^{13} - q^{14} + q^{16} + 2q^{17} - 3q^{18} + O(q^{20})\)  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.