Properties

Label 5040o
Number of curves $4$
Conductor $5040$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 5040o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.x4 5040o1 \([0, 0, 0, 78, -2189]\) \(4499456/180075\) \(-2100394800\) \([2]\) \(2048\) \(0.46863\) \(\Gamma_0(N)\)-optimal
5040.x3 5040o2 \([0, 0, 0, -2127, -36146]\) \(5702413264/275625\) \(51438240000\) \([2, 2]\) \(4096\) \(0.81521\)  
5040.x1 5040o3 \([0, 0, 0, -33627, -2373446]\) \(5633270409316/14175\) \(10581580800\) \([2]\) \(8192\) \(1.1618\)  
5040.x2 5040o4 \([0, 0, 0, -5907, 127906]\) \(30534944836/8203125\) \(6123600000000\) \([4]\) \(8192\) \(1.1618\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5040o have rank \(1\).

Complex multiplication

The elliptic curves in class 5040o do not have complex multiplication.

Modular form 5040.2.a.o

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.