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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 451770.2.a.d

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

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
451770.d1 451770d2 \([1, 1, 0, -4653, 118503]\) \(220020692653/3608550\) \(182783883150\) \([2]\) \(645120\) \(0.96000\) \(\Gamma_0(N)\)-optimal*
451770.d2 451770d1 \([1, 1, 0, -583, -2783]\) \(433798093/196020\) \(9929001060\) \([2]\) \(322560\) \(0.61343\) \(\Gamma_0(N)\)-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.d1.