Properties

Label 405042.dm
Number of curves $2$
Conductor $405042$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("dm1")
 
E.isogeny_class()
 

Elliptic curves in class 405042.dm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405042.dm1 405042dm2 \([1, 0, 0, -1077412, 429259760]\) \(2940001530995593/8673562656\) \(408055396560219936\) \([2]\) \(6842880\) \(2.2492\) \(\Gamma_0(N)\)-optimal*
405042.dm2 405042dm1 \([1, 0, 0, -95492, 553488]\) \(2046931732873/1181672448\) \(55592821369586688\) \([2]\) \(3421440\) \(1.9026\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 405042.dm1.

Rank

sage: E.rank()
 

The elliptic curves in class 405042.dm have rank \(0\).

Complex multiplication

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

Modular form 405042.2.a.dm

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.