# Properties

 Label 31200cg Number of curves $4$ Conductor $31200$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("cg1")

sage: E.isogeny_class()

## Elliptic curves in class 31200cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.cj3 31200cg1 $$[0, 1, 0, -2158, 25688]$$ $$1111934656/342225$$ $$342225000000$$ $$[2, 2]$$ $$36864$$ $$0.91789$$ $$\Gamma_0(N)$$-optimal
31200.cj4 31200cg2 $$[0, 1, 0, 5967, 180063]$$ $$367061696/426465$$ $$-27293760000000$$ $$[4]$$ $$73728$$ $$1.2645$$
31200.cj2 31200cg3 $$[0, 1, 0, -13408, -581812]$$ $$33324076232/1285245$$ $$10281960000000$$ $$[2]$$ $$73728$$ $$1.2645$$
31200.cj1 31200cg4 $$[0, 1, 0, -31408, 2131688]$$ $$428320044872/73125$$ $$585000000000$$ $$[2]$$ $$73728$$ $$1.2645$$

## Rank

sage: E.rank()

The elliptic curves in class 31200cg have rank $$0$$.

## Complex multiplication

The elliptic curves in class 31200cg do not have complex multiplication.

## Modular form 31200.2.a.cg

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.