# Properties

 Label 187200.hz Number of curves $4$ Conductor $187200$ CM no Rank $2$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 187200.hz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
187200.hz1 187200ml4 $$[0, 0, 0, -194700, -26894000]$$ $$2186875592/428415$$ $$159905041920000000$$ $$[2]$$ $$1179648$$ $$2.0178$$
187200.hz2 187200ml2 $$[0, 0, 0, -59700, 5236000]$$ $$504358336/38025$$ $$1774094400000000$$ $$[2, 2]$$ $$589824$$ $$1.6712$$
187200.hz3 187200ml1 $$[0, 0, 0, -58575, 5456500]$$ $$30488290624/195$$ $$142155000000$$ $$[2]$$ $$294912$$ $$1.3247$$ $$\Gamma_0(N)$$-optimal
187200.hz4 187200ml3 $$[0, 0, 0, 57300, 23254000]$$ $$55742968/658125$$ $$-245643840000000000$$ $$[2]$$ $$1179648$$ $$2.0178$$

## Rank

sage: E.rank()

The elliptic curves in class 187200.hz have rank $$2$$.

## Complex multiplication

The elliptic curves in class 187200.hz do not have complex multiplication.

## Modular form 187200.2.a.hz

sage: E.q_eigenform(10)

$$q - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.