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Properties

Label 438702.b
Number of curves $2$
Conductor $438702$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 438702.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
438702.b1 438702b2 \([1, 1, 0, -125304, -17124696]\) \(752880570738483817/2933705016\) \(847840749624\) \([]\) \(1990656\) \(1.5018\)  
438702.b2 438702b1 \([1, 1, 0, -2139, -4761]\) \(3747791557657/2146477806\) \(620332085934\) \([]\) \(663552\) \(0.95249\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 438702.b1.

Rank

sage: E.rank()
 

The elliptic curves in class 438702.b have rank \(2\).

Complex multiplication

The elliptic curves in class 438702.b do not have complex multiplication.

Modular form 438702.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} + q^{11} - q^{12} + 2 q^{13} - q^{14} + 3 q^{15} + q^{16} - q^{18} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.