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Properties

Label 70070f
Number of curves $2$
Conductor $70070$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 70070f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
70070.c2 70070f1 \([1, 0, 1, -6739, -219474]\) \(-287626699801/9518080\) \(-1119792593920\) \([2]\) \(138240\) \(1.0867\) \(\Gamma_0(N)\)-optimal
70070.c1 70070f2 \([1, 0, 1, -108659, -13795218]\) \(1205943158724121/1258400\) \(148049501600\) \([2]\) \(276480\) \(1.4333\)  

Rank

sage: E.rank()
 

The elliptic curves in class 70070f have rank \(1\).

Complex multiplication

The elliptic curves in class 70070f do not have complex multiplication.

Modular form 70070.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} - q^{8} + q^{9} + q^{10} - q^{11} - 2 q^{12} + q^{13} + 2 q^{15} + q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) Copy content 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 Cremona 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 Cremona labels.