Properties

Label 130050.fm
Number of curves $6$
Conductor $130050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fm1")
 
E.isogeny_class()
 

Elliptic curves in class 130050.fm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
130050.fm1 130050br5 \([1, -1, 1, -1804054955, 29493709544297]\) \(2361739090258884097/5202\) \(1430248267825031250\) \([2]\) \(37748736\) \(3.6177\)  
130050.fm2 130050br3 \([1, -1, 1, -112754705, 460849452797]\) \(576615941610337/27060804\) \(7440151489225812562500\) \([2, 2]\) \(18874368\) \(3.2711\)  
130050.fm3 130050br6 \([1, -1, 1, -106902455, 510815963297]\) \(-491411892194497/125563633938\) \(-34522716251757171724031250\) \([2]\) \(37748736\) \(3.6177\)  
130050.fm4 130050br2 \([1, -1, 1, -7414205, 6410535797]\) \(163936758817/30338064\) \(8341207897955582250000\) \([2, 2]\) \(9437184\) \(2.9246\)  
130050.fm5 130050br1 \([1, -1, 1, -2212205, -1173980203]\) \(4354703137/352512\) \(96920353207908000000\) \([2]\) \(4718592\) \(2.5780\) \(\Gamma_0(N)\)-optimal
130050.fm6 130050br4 \([1, -1, 1, 14694295, 37318218797]\) \(1276229915423/2927177028\) \(-804803897330684928562500\) \([2]\) \(18874368\) \(3.2711\)  

Rank

sage: E.rank()
 

The elliptic curves in class 130050.fm have rank \(1\).

Complex multiplication

The elliptic curves in class 130050.fm do not have complex multiplication.

Modular form 130050.2.a.fm

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} - 4 q^{11} + 2 q^{13} + q^{16} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.