# Properties

 Label 4998.bp Number of curves $4$ Conductor $4998$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bp1")

sage: E.isogeny_class()

## Elliptic curves in class 4998.bp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4998.bp1 4998bl3 $$[1, 0, 0, -248872, -47807992]$$ $$14489843500598257/6246072$$ $$734844124728$$ $$$$ $$36864$$ $$1.6195$$
4998.bp2 4998bl4 $$[1, 0, 0, -33272, 1231992]$$ $$34623662831857/14438442312$$ $$1698668299564488$$ $$$$ $$36864$$ $$1.6195$$
4998.bp3 4998bl2 $$[1, 0, 0, -15632, -740160]$$ $$3590714269297/73410624$$ $$8636686502976$$ $$[2, 2]$$ $$18432$$ $$1.2729$$
4998.bp4 4998bl1 $$[1, 0, 0, 48, -34560]$$ $$103823/4386816$$ $$-516104515584$$ $$$$ $$9216$$ $$0.92636$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4998.bp have rank $$0$$.

## Complex multiplication

The elliptic curves in class 4998.bp do not have complex multiplication.

## Modular form4998.2.a.bp

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} + q^{12} + 6q^{13} + 2q^{15} + q^{16} - q^{17} + q^{18} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 