Properties

Label 450840.x
Number of curves $4$
Conductor $450840$
CM no
Rank $1$
Graph

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

Elliptic curves in class 450840.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.x1 450840x4 \([0, -1, 0, -18061440, -29538245700]\) \(26362547147244676/244298925\) \(6038304930624844800\) \([2]\) \(21233664\) \(2.7674\)  
450840.x2 450840x2 \([0, -1, 0, -1154940, -438777900]\) \(27572037674704/2472575625\) \(15278582977575840000\) \([2, 2]\) \(10616832\) \(2.4208\)  
450840.x3 450840x1 \([0, -1, 0, -251815, 40962100]\) \(4572531595264/776953125\) \(300060154631250000\) \([4]\) \(5308416\) \(2.0742\) \(\Gamma_0(N)\)-optimal*
450840.x4 450840x3 \([0, -1, 0, 1301560, -2057120100]\) \(9865576607324/79640206425\) \(-1968456681223865164800\) \([2]\) \(21233664\) \(2.7674\)  
*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 450840.x1.

Rank

sage: E.rank()
 

The elliptic curves in class 450840.x have rank \(1\).

Complex multiplication

The elliptic curves in class 450840.x do not have complex multiplication.

Modular form 450840.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} - 4q^{11} + q^{13} - q^{15} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.