Properties

Label 162450dq
Number of curves $4$
Conductor $162450$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 162450dq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162450.w3 162450dq1 \([1, -1, 0, -2519667, 1530856741]\) \(3301293169/22800\) \(12218109332456250000\) \([2]\) \(4423680\) \(2.4960\) \(\Gamma_0(N)\)-optimal
162450.w2 162450dq2 \([1, -1, 0, -4144167, -683336759]\) \(14688124849/8122500\) \(4352701449687539062500\) \([2, 2]\) \(8847360\) \(2.8426\)  
162450.w4 162450dq3 \([1, -1, 0, 16162083, -5414693009]\) \(871257511151/527800050\) \(-282838540200696288281250\) \([2]\) \(17694720\) \(3.1892\)  
162450.w1 162450dq4 \([1, -1, 0, -50442417, -137679858509]\) \(26487576322129/44531250\) \(23863494789953613281250\) \([2]\) \(17694720\) \(3.1892\)  

Rank

sage: E.rank()
 

The elliptic curves in class 162450dq have rank \(1\).

Complex multiplication

The elliptic curves in class 162450dq do not have complex multiplication.

Modular form 162450.2.a.dq

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