Properties

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

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 5040.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.d1 5040bd3 \([0, 0, 0, -16203, -793798]\) \(157551496201/13125\) \(39191040000\) \([2]\) \(8192\) \(1.0765\)  
5040.d2 5040bd2 \([0, 0, 0, -1083, -10582]\) \(47045881/11025\) \(32920473600\) \([2, 2]\) \(4096\) \(0.72992\)  
5040.d3 5040bd1 \([0, 0, 0, -363, 2522]\) \(1771561/105\) \(313528320\) \([2]\) \(2048\) \(0.38334\) \(\Gamma_0(N)\)-optimal
5040.d4 5040bd4 \([0, 0, 0, 2517, -66022]\) \(590589719/972405\) \(-2903585771520\) \([2]\) \(8192\) \(1.0765\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 5040.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} - 6 q^{13} - 2 q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.