Properties

Label 139650.ef
Number of curves $6$
Conductor $139650$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ef1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 139650.ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.ef1 139650ha5 \([1, 0, 1, -435505901, 3497436384698]\) \(4969327007303723277361/1123462695162150\) \(2065222853486434146093750\) \([2]\) \(70778880\) \(3.6579\)  
139650.ef2 139650ha3 \([1, 0, 1, -30337151, 41346947198]\) \(1679731262160129361/570261564022500\) \(1048292230401298476562500\) \([2, 2]\) \(35389440\) \(3.3113\)  
139650.ef3 139650ha2 \([1, 0, 1, -12476651, -16485351802]\) \(116844823575501841/3760263939600\) \(6912363941093756250000\) \([2, 2]\) \(17694720\) \(2.9647\)  
139650.ef4 139650ha1 \([1, 0, 1, -12378651, -16764259802]\) \(114113060120923921/124104960\) \(228137881860000000\) \([2]\) \(8847360\) \(2.6181\) \(\Gamma_0(N)\)-optimal
139650.ef5 139650ha4 \([1, 0, 1, 3815849, -56467146802]\) \(3342636501165359/751262567039460\) \(-1381020152337897336562500\) \([2]\) \(35389440\) \(3.3113\)  
139650.ef6 139650ha6 \([1, 0, 1, 89063599, 286596087698]\) \(42502666283088696719/43898058864843750\) \(-80696292615468786621093750\) \([2]\) \(70778880\) \(3.6579\)  

Rank

sage: E.rank()
 

The elliptic curves in class 139650.ef have rank \(0\).

Complex multiplication

The elliptic curves in class 139650.ef do not have complex multiplication.

Modular form 139650.2.a.ef

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.