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Properties

Label 451770.x
Number of curves $2$
Conductor $451770$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("x1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 451770.x have rank \(1\).

Complex multiplication

The elliptic curves in class 451770.x do not have complex multiplication.

Modular form 451770.2.a.x

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 2 q^{13} + 2 q^{14} + q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 451770.x

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
451770.x1 451770x2 \([1, 0, 1, -16013, -99844]\) \(8963913629917/5104687500\) \(258567735937500\) \([2]\) \(1824768\) \(1.4553\) \(\Gamma_0(N)\)-optimal*
451770.x2 451770x1 \([1, 0, 1, 3967, -11932]\) \(136352311523/80190000\) \(-4061864070000\) \([2]\) \(912384\) \(1.1088\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 451770.x1.