Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.a4 450840a1 \([0, -1, 0, -663351, -175098924]\) \(83587439220736/13990184325\) \(5403024631478494800\) \([2]\) \(7864320\) \(2.3156\) \(\Gamma_0(N)\)-optimal
450840.a2 450840a2 \([0, -1, 0, -10143996, -12431676780]\) \(18681746265374416/693005625\) \(4282232599251360000\) \([2, 2]\) \(15728640\) \(2.6621\)  
450840.a3 450840a3 \([0, -1, 0, -9675816, -13631528484]\) \(-4053153720264484/903687890625\) \(-22336336705971600000000\) \([2]\) \(31457280\) \(3.0087\)  
450840.a1 450840a4 \([0, -1, 0, -162302496, -795804498180]\) \(19129597231400697604/26325\) \(650671620019200\) \([2]\) \(31457280\) \(3.0087\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 450840.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - 4q^{7} + q^{9} + q^{13} + q^{15} + O(q^{20})\)  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.