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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("y1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 451770.2.a.y

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 451770.y

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
451770.y1 451770y2 \([1, 0, 1, -1295103, -146756744]\) \(93632326352929/50564357250\) \(129734306750435615250\) \([]\) \(23639040\) \(2.5504\)  
451770.y2 451770y1 \([1, 0, 1, -761193, 255548548]\) \(19010647320769/439560\) \(1127790700340040\) \([]\) \(7879680\) \(2.0011\) \(\Gamma_0(N)\)-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 451770.y1.