# Properties

 Label 190575.dx Number of curves $6$ Conductor $190575$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dx1")

sage: E.isogeny_class()

## Elliptic curves in class 190575.dx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
190575.dx1 190575do6 $$[1, -1, 0, -83014470567, -9206145580482284]$$ $$3135316978843283198764801/571725$$ $$11536945696508203125$$ $$[2]$$ $$176947200$$ $$4.4521$$
190575.dx2 190575do4 $$[1, -1, 0, -5188404942, -143845020910409]$$ $$765458482133960722801/326869475625$$ $$6595960278336152431640625$$ $$[2, 2]$$ $$88473600$$ $$4.1055$$
190575.dx3 190575do5 $$[1, -1, 0, -5162677317, -145342188592034]$$ $$-754127868744065783521/15825714261328125$$ $$-319350047735183888067626953125$$ $$[2]$$ $$176947200$$ $$4.4521$$
190575.dx4 190575do3 $$[1, -1, 0, -692740692, 3701012243341]$$ $$1821931919215868881/761147600816295$$ $$15359339783366300714919609375$$ $$[2]$$ $$88473600$$ $$4.1055$$
190575.dx5 190575do2 $$[1, -1, 0, -325883817, -2224093144784]$$ $$189674274234120481/3859869269025$$ $$77889023835519240003515625$$ $$[2, 2]$$ $$44236800$$ $$3.7590$$
190575.dx6 190575do1 $$[1, -1, 0, 952308, -103907201909]$$ $$4733169839/231139696095$$ $$-4664211154235732516484375$$ $$[2]$$ $$22118400$$ $$3.4124$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 190575.dx have rank $$0$$.

## Complex multiplication

The elliptic curves in class 190575.dx do not have complex multiplication.

## Modular form 190575.2.a.dx

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{7} - 3q^{8} - 2q^{13} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.