# Properties

 Label 212160.gx Number of curves $4$ Conductor $212160$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("gx1")

sage: E.isogeny_class()

## Elliptic curves in class 212160.gx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.gx1 212160r3 $$[0, 1, 0, -249985, 48024575]$$ $$26362547147244676/244298925$$ $$16010374348800$$ $$[2]$$ $$1179648$$ $$1.6974$$
212160.gx2 212160r2 $$[0, 1, 0, -15985, 709775]$$ $$27572037674704/2472575625$$ $$40510679040000$$ $$[2, 2]$$ $$589824$$ $$1.3508$$
212160.gx3 212160r1 $$[0, 1, 0, -3485, -67725]$$ $$4572531595264/776953125$$ $$795600000000$$ $$[2]$$ $$294912$$ $$1.0042$$ $$\Gamma_0(N)$$-optimal
212160.gx4 212160r4 $$[0, 1, 0, 18015, 3354975]$$ $$9865576607324/79640206425$$ $$-5219300568268800$$ $$[2]$$ $$1179648$$ $$1.6974$$

## Rank

sage: E.rank()

The elliptic curves in class 212160.gx have rank $$1$$.

## Complex multiplication

The elliptic curves in class 212160.gx do not have complex multiplication.

## Modular form 212160.2.a.gx

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 4q^{11} - q^{13} + q^{15} + q^{17} - 8q^{19} + O(q^{20})$$

## 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.