Properties

Label 47040.dp
Number of curves $4$
Conductor $47040$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("dp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040.dp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.dp1 47040fi4 \([0, -1, 0, -219585, 39678465]\) \(303735479048/105\) \(404787855360\) \([4]\) \(294912\) \(1.5831\)  
47040.dp2 47040fi2 \([0, -1, 0, -13785, 617625]\) \(601211584/11025\) \(5312840601600\) \([2, 2]\) \(147456\) \(1.2365\)  
47040.dp3 47040fi1 \([0, -1, 0, -1780, -13838]\) \(82881856/36015\) \(271176239040\) \([2]\) \(73728\) \(0.88997\) \(\Gamma_0(N)\)-optimal
47040.dp4 47040fi3 \([0, -1, 0, -65, 1778337]\) \(-8/354375\) \(-1366159011840000\) \([2]\) \(294912\) \(1.5831\)  

Rank

sage: E.rank()
 

The elliptic curves in class 47040.dp have rank \(0\).

Complex multiplication

The elliptic curves in class 47040.dp do not have complex multiplication.

Modular form 47040.2.a.dp

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + 4q^{11} + 6q^{13} - q^{15} + 6q^{17} - 4q^{19} + O(q^{20})\)  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 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.