# Properties

 Label 117600.hz Number of curves $4$ Conductor $117600$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 117600.hz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117600.hz1 117600dd4 $$[0, 1, 0, -1372408, -619289812]$$ $$303735479048/105$$ $$98825160000000$$ $$[2]$$ $$1769472$$ $$2.0413$$
117600.hz2 117600dd3 $$[0, 1, 0, -178033, 14372063]$$ $$82881856/36015$$ $$271176239040000000$$ $$[2]$$ $$1769472$$ $$2.0413$$
117600.hz3 117600dd1 $$[0, 1, 0, -86158, -9607312]$$ $$601211584/11025$$ $$1297080225000000$$ $$[2, 2]$$ $$884736$$ $$1.6947$$ $$\Gamma_0(N)$$-optimal
117600.hz4 117600dd2 $$[0, 1, 0, -408, -27786312]$$ $$-8/354375$$ $$-333534915000000000$$ $$[2]$$ $$1769472$$ $$2.0413$$

## Rank

sage: E.rank()

The elliptic curves in class 117600.hz have rank $$1$$.

## Complex multiplication

The elliptic curves in class 117600.hz do not have complex multiplication.

## Modular form 117600.2.a.hz

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} + 4q^{11} + 6q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.