Properties

Label 446160.dw
Number of curves $4$
Conductor $446160$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("dw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 446160.dw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.dw1 446160dw4 \([0, -1, 0, -42249380, 105714969372]\) \(6749703004355978704/5671875\) \(7008526668000000\) \([2]\) \(15925248\) \(2.7760\) \(\Gamma_0(N)\)-optimal*
446160.dw2 446160dw3 \([0, -1, 0, -2640005, 1653219372]\) \(-26348629355659264/24169921875\) \(-1866617542968750000\) \([2]\) \(7962624\) \(2.4295\) \(\Gamma_0(N)\)-optimal*
446160.dw3 446160dw2 \([0, -1, 0, -533420, 138254700]\) \(13584145739344/1195803675\) \(1477610480825107200\) \([2]\) \(5308416\) \(2.2267\) \(\Gamma_0(N)\)-optimal*
446160.dw4 446160dw1 \([0, -1, 0, 36955, 10034400]\) \(72268906496/606436875\) \(-46834479458910000\) \([2]\) \(2654208\) \(1.8801\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 446160.dw1.

Rank

sage: E.rank()
 

The elliptic curves in class 446160.dw have rank \(0\).

Complex multiplication

The elliptic curves in class 446160.dw do not have complex multiplication.

Modular form 446160.2.a.dw

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + 2q^{7} + q^{9} + q^{11} - q^{15} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.