Properties

Label 41650p
Number of curves $4$
Conductor $41650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 41650p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41650.s3 41650p1 \([1, -1, 0, -22892, -1305984]\) \(721734273/13328\) \(24500404250000\) \([2]\) \(98304\) \(1.3637\) \(\Gamma_0(N)\)-optimal
41650.s2 41650p2 \([1, -1, 0, -47392, 2001516]\) \(6403769793/2775556\) \(5102209185062500\) \([2, 2]\) \(196608\) \(1.7102\)  
41650.s4 41650p3 \([1, -1, 0, 160858, 14704766]\) \(250404380127/196003234\) \(-360306007451031250\) \([2]\) \(393216\) \(2.0568\)  
41650.s1 41650p4 \([1, -1, 0, -647642, 200684266]\) \(16342588257633/8185058\) \(15046310760031250\) \([2]\) \(393216\) \(2.0568\)  

Rank

sage: E.rank()
 

The elliptic curves in class 41650p have rank \(1\).

Complex multiplication

The elliptic curves in class 41650p do not have complex multiplication.

Modular form 41650.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 3 q^{9} - 2 q^{13} + q^{16} + q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.