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Properties

Label 41209.d
Number of curves $2$
Conductor $41209$
CM no
Rank $0$
Graph

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

Elliptic curves in class 41209.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41209.d1 41209f2 \([1, 1, 1, -6367649, -6187308254]\) \(408023180713/1421\) \(99442104195999509\) \([2]\) \(967680\) \(2.4810\)  
41209.d2 41209f1 \([1, 1, 1, -392344, -99667520]\) \(-95443993/5887\) \(-411974431669140823\) \([2]\) \(483840\) \(2.1344\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 41209.d have rank \(0\).

Complex multiplication

The elliptic curves in class 41209.d do not have complex multiplication.

Modular form 41209.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - 2 q^{12} + 2 q^{13} - 4 q^{15} - q^{16} + 4 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.