Properties

Label 129360hz
Number of curves $4$
Conductor $129360$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("hz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 129360hz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.hl3 129360hz1 \([0, 1, 0, -130405, 18350618]\) \(-130287139815424/2250652635\) \(-4236592509681840\) \([2]\) \(995328\) \(1.7965\) \(\Gamma_0(N)\)-optimal
129360.hl2 129360hz2 \([0, 1, 0, -2095060, 1166495000]\) \(33766427105425744/9823275\) \(295858811001600\) \([2]\) \(1990656\) \(2.1431\)  
129360.hl4 129360hz3 \([0, 1, 0, 504635, 88316150]\) \(7549996227362816/6152409907875\) \(-11581197972025374000\) \([2]\) \(2985984\) \(2.3458\)  
129360.hl1 129360hz4 \([0, 1, 0, -2430220, 768028568]\) \(52702650535889104/22020583921875\) \(663219117523116000000\) \([2]\) \(5971968\) \(2.6924\)  

Rank

sage: E.rank()
 

The elliptic curves in class 129360hz have rank \(0\).

Complex multiplication

The elliptic curves in class 129360hz do not have complex multiplication.

Modular form 129360.2.a.hz

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} + q^{11} - 2q^{13} + q^{15} + 6q^{17} + 8q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.