Properties

Label 117208.m
Number of curves $4$
Conductor $117208$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 117208.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117208.m1 117208d4 \([0, 0, 0, -2187899, -1245629882]\) \(9614292367656708/2093\) \(252149101568\) \([2]\) \(786432\) \(2.0121\)  
117208.m2 117208d3 \([0, 0, 0, -159299, -12610738]\) \(3710860803108/1577224103\) \(190012250617699328\) \([2]\) \(786432\) \(2.0121\)  
117208.m3 117208d2 \([0, 0, 0, -136759, -19458390]\) \(9392111857872/4380649\) \(131937017395456\) \([2, 2]\) \(393216\) \(1.6656\)  
117208.m4 117208d1 \([0, 0, 0, -7154, -406455]\) \(-21511084032/25465531\) \(-47935908105904\) \([2]\) \(196608\) \(1.3190\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 117208.m have rank \(1\).

Complex multiplication

The elliptic curves in class 117208.m do not have complex multiplication.

Modular form 117208.2.a.m

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.