Properties

Label 140679.z
Number of curves $4$
Conductor $140679$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 140679.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
140679.z1 140679y3 \([1, -1, 0, -23529858, 43932670109]\) \(16798320881842096017/2132227789307\) \(182872906577266668147\) \([2]\) \(8257536\) \(2.9098\)  
140679.z2 140679y4 \([1, -1, 0, -9334068, -10525526425]\) \(1048626554636928177/48569076788309\) \(4165581316684401079389\) \([2]\) \(8257536\) \(2.9098\)  
140679.z3 140679y2 \([1, -1, 0, -1596723, 562088960]\) \(5249244962308257/1448621666569\) \(124242661138178508849\) \([2, 2]\) \(4128768\) \(2.5632\)  
140679.z4 140679y1 \([1, -1, 0, 257682, 57319919]\) \(22062729659823/29354283343\) \(-2517603017064022503\) \([2]\) \(2064384\) \(2.2166\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 140679.z have rank \(0\).

Complex multiplication

The elliptic curves in class 140679.z do not have complex multiplication.

Modular form 140679.2.a.z

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