# Properties

 Label 193200.hi Number of curves $4$ Conductor $193200$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("hi1")

sage: E.isogeny_class()

## Elliptic curves in class 193200.hi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.hi1 193200fs3 $$[0, 1, 0, -4637008, 3841759988]$$ $$344577854816148242/2716875$$ $$86940000000000$$ $$[2]$$ $$2949120$$ $$2.2656$$
193200.hi2 193200fs2 $$[0, 1, 0, -290008, 59869988]$$ $$168591300897604/472410225$$ $$7558563600000000$$ $$[2, 2]$$ $$1474560$$ $$1.9191$$
193200.hi3 193200fs4 $$[0, 1, 0, -175008, 107939988]$$ $$-18524646126002/146738831715$$ $$-4695642614880000000$$ $$[4]$$ $$2949120$$ $$2.2656$$
193200.hi4 193200fs1 $$[0, 1, 0, -25508, 92988]$$ $$458891455696/264449745$$ $$1057798980000000$$ $$[2]$$ $$737280$$ $$1.5725$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 193200.hi have rank $$1$$.

## Complex multiplication

The elliptic curves in class 193200.hi do not have complex multiplication.

## Modular form 193200.2.a.hi

sage: E.q_eigenform(10)

$$q + q^{3} + q^{7} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.