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Properties

Label 405042.dp
Number of curves $4$
Conductor $405042$
CM no
Rank $1$
Graph

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

Elliptic curves in class 405042.dp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405042.dp1 405042dp4 \([1, 0, 0, -174702167, -888777840483]\) \(12534210458299016895673/315581882565708\) \(14846827692942273248748\) \([2]\) \(106168320\) \(3.3625\)  
405042.dp2 405042dp2 \([1, 0, 0, -11335227, -12771634815]\) \(3423676911662954233/483711578981136\) \(22756637383068625500816\) \([2, 2]\) \(53084160\) \(3.0159\)  
405042.dp3 405042dp1 \([1, 0, 0, -2988907, 1789355057]\) \(62768149033310713/6915442583808\) \(325343088860163694848\) \([2]\) \(26542080\) \(2.6693\) \(\Gamma_0(N)\)-optimal*
405042.dp4 405042dp3 \([1, 0, 0, 18490593, -68635395675]\) \(14861225463775641287/51859390496937804\) \(-2439770714051466791385324\) \([2]\) \(106168320\) \(3.3625\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405042.dp1.

Rank

sage: E.rank()
 

The elliptic curves in class 405042.dp have rank \(1\).

Complex multiplication

The elliptic curves in class 405042.dp do not have complex multiplication.

Modular form 405042.2.a.dp

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{15} + q^{16} - q^{17} + q^{18} + O(q^{20})\) 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.