# Properties

 Label 129360.gc Number of curves $4$ Conductor $129360$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gc1")

sage: E.isogeny_class()

## Elliptic curves in class 129360.gc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.gc1 129360hn4 $$[0, 1, 0, -12249820, -16506315400]$$ $$6749703004355978704/5671875$$ $$170826348000000$$ $$[2]$$ $$2488320$$ $$2.4665$$
129360.gc2 129360hn3 $$[0, 1, 0, -765445, -258221650]$$ $$-26348629355659264/24169921875$$ $$-45497074218750000$$ $$[2]$$ $$1244160$$ $$2.1199$$
129360.gc3 129360hn2 $$[0, 1, 0, -154660, -21608392]$$ $$13584145739344/1195803675$$ $$36015387279379200$$ $$[2]$$ $$829440$$ $$1.9172$$
129360.gc4 129360hn1 $$[0, 1, 0, 10715, -1564942]$$ $$72268906496/606436875$$ $$-1141547070510000$$ $$[2]$$ $$414720$$ $$1.5706$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 129360.gc have rank $$1$$.

## Complex multiplication

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

## Modular form 129360.2.a.gc

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.