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Properties

Label 422370.dw
Number of curves $6$
Conductor $422370$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("dw1")
 
E.isogeny_class()
 

Elliptic curves in class 422370.dw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422370.dw1 422370dw6 \([1, -1, 1, -29289803, -61005726943]\) \(81025909800741361/11088090\) \(380282093777164410\) \([2]\) \(28311552\) \(2.7856\)  
422370.dw2 422370dw3 \([1, -1, 1, -2745473, 1750254581]\) \(66730743078481/60937500\) \(2089939754235937500\) \([2]\) \(14155776\) \(2.4390\) \(\Gamma_0(N)\)-optimal*
422370.dw3 422370dw4 \([1, -1, 1, -1835753, -947247163]\) \(19948814692561/231344100\) \(7934280722017380900\) \([2, 2]\) \(14155776\) \(2.4390\)  
422370.dw4 422370dw5 \([1, -1, 1, -373703, -2415730183]\) \(-168288035761/73415764890\) \(-2517899907794871287610\) \([2]\) \(28311552\) \(2.7856\)  
422370.dw5 422370dw2 \([1, -1, 1, -211253, 13807037]\) \(30400540561/15210000\) \(521648962657290000\) \([2, 2]\) \(7077888\) \(2.0925\) \(\Gamma_0(N)\)-optimal*
422370.dw6 422370dw1 \([1, -1, 1, 48667, 1642781]\) \(371694959/249600\) \(-8560393233350400\) \([2]\) \(3538944\) \(1.7459\) \(\Gamma_0(N)\)-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 422370.dw1.

Rank

sage: E.rank()
 

The elliptic curves in class 422370.dw have rank \(0\).

Complex multiplication

The elliptic curves in class 422370.dw do not have complex multiplication.

Modular form 422370.2.a.dw

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.