# Properties

 Label 4368.j Number of curves $4$ Conductor $4368$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("j1")

E.isogeny_class()

## Elliptic curves in class 4368.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4368.j1 4368b3 $$[0, -1, 0, -54192, 757152]$$ $$8594236719188066/4858291807551$$ $$9949781621864448$$ $$$$ $$32256$$ $$1.7597$$
4368.j2 4368b2 $$[0, -1, 0, -40152, 3104640]$$ $$6991270724335972/14494474449$$ $$14842341835776$$ $$[2, 2]$$ $$16128$$ $$1.4131$$
4368.j3 4368b1 $$[0, -1, 0, -40132, 3107872]$$ $$27923315228972368/120393$$ $$30820608$$ $$$$ $$8064$$ $$1.0665$$ $$\Gamma_0(N)$$-optimal
4368.j4 4368b4 $$[0, -1, 0, -26432, 5244960]$$ $$-997241325462146/5206220835543$$ $$-10662340271192064$$ $$$$ $$32256$$ $$1.7597$$

## Rank

sage: E.rank()

The elliptic curves in class 4368.j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4368.j do not have complex multiplication.

## Modular form4368.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} + 2 q^{5} - q^{7} + q^{9} - 4 q^{11} - q^{13} - 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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. 