# Properties

 Label 193200.hl Number of curves $6$ Conductor $193200$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 193200.hl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.hl1 193200ft5 $$[0, 1, 0, -35554008, 81586415988]$$ $$155324313723954725282/13018359375$$ $$416587500000000000$$ $$[2]$$ $$11010048$$ $$2.8234$$
193200.hl2 193200ft4 $$[0, 1, 0, -3060008, -2059146012]$$ $$198048499826486404/242568272835$$ $$3881092365360000000$$ $$[2]$$ $$5505024$$ $$2.4768$$
193200.hl3 193200ft3 $$[0, 1, 0, -2227008, 1268345988]$$ $$76343005935514084/694180580625$$ $$11106889290000000000$$ $$[2, 2]$$ $$5505024$$ $$2.4768$$
193200.hl4 193200ft6 $$[0, 1, 0, -652008, 3029195988]$$ $$-957928673903042/123339801817575$$ $$-3946873658162400000000$$ $$[2]$$ $$11010048$$ $$2.8234$$
193200.hl5 193200ft2 $$[0, 1, 0, -242508, -13641012]$$ $$394315384276816/208332909225$$ $$833331636900000000$$ $$[2, 2]$$ $$2752512$$ $$2.1302$$
193200.hl6 193200ft1 $$[0, 1, 0, 57617, -1636012]$$ $$84611246065664/53699121315$$ $$-13424780328750000$$ $$[2]$$ $$1376256$$ $$1.7837$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 193200.2.a.hl

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.