# Properties

 Label 8050p Number of curves $2$ Conductor $8050$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 8050p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8050.n2 8050p1 $$[1, 0, 0, 862, 45892]$$ $$4533086375/60669952$$ $$-947968000000$$ $$$$ $$16128$$ $$0.97890$$ $$\Gamma_0(N)$$-optimal
8050.n1 8050p2 $$[1, 0, 0, -15138, 669892]$$ $$24553362849625/1755162752$$ $$27424418000000$$ $$$$ $$32256$$ $$1.3255$$

## Rank

sage: E.rank()

The elliptic curves in class 8050p have rank $$1$$.

## Complex multiplication

The elliptic curves in class 8050p do not have complex multiplication.

## Modular form8050.2.a.p

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + q^{8} + q^{9} + 4q^{11} - 2q^{12} - q^{14} + q^{16} - 6q^{17} + q^{18} - 6q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 