# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hl1")

sage: E.isogeny_class()

## Elliptic curves in class 129360.hl

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 129360.hl have rank $$0$$.

## Complex multiplication

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

## Modular form 129360.2.a.hl

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + q^{11} - 2q^{13} + q^{15} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.