Properties

Label 450840.w
Number of curves $4$
Conductor $450840$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.w1 450840w4 \([0, -1, 0, -9449240, -11176780980]\) \(1887517194957938/21849165\) \(1080085970493204480\) \([2]\) \(14155776\) \(2.6115\)  
450840.w2 450840w2 \([0, -1, 0, -605840, -164979300]\) \(994958062276/98903025\) \(2444573276412134400\) \([2, 2]\) \(7077888\) \(2.2649\)  
450840.w3 450840w1 \([0, -1, 0, -137660, 16861812]\) \(46689225424/7249905\) \(44798741038321920\) \([2]\) \(3538944\) \(1.9183\) \(\Gamma_0(N)\)-optimal*
450840.w4 450840w3 \([0, -1, 0, 746680, -798499668]\) \(931329171502/6107473125\) \(-301915246531242240000\) \([2]\) \(14155776\) \(2.6115\)  
*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.w1.

Rank

sage: E.rank()
 

The elliptic curves in class 450840.w have rank \(0\).

Complex multiplication

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

Modular form 450840.2.a.w

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.