# Properties

 Label 51600.bg Number of curves $2$ Conductor $51600$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("bg1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.bg1 51600cf1 $$[0, -1, 0, -3208, -115088]$$ $$-2282665/2322$$ $$-3715200000000$$ $$[]$$ $$86400$$ $$1.1075$$ $$\Gamma_0(N)$$-optimal
51600.bg2 51600cf2 $$[0, -1, 0, 26792, 2044912]$$ $$1329238535/1908168$$ $$-3053068800000000$$ $$[]$$ $$259200$$ $$1.6568$$

## Rank

sage: E.rank()

The elliptic curves in class 51600.bg have rank $$1$$.

## Complex multiplication

The elliptic curves in class 51600.bg do not have complex multiplication.

## Modular form 51600.2.a.bg

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 