# Properties

 Label 10080s Number of curves $4$ Conductor $10080$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 10080s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10080.bc3 10080s1 $$[0, 0, 0, -633, 6032]$$ $$601211584/11025$$ $$514382400$$ $$[2, 2]$$ $$6144$$ $$0.46632$$ $$\Gamma_0(N)$$-optimal
10080.bc2 10080s2 $$[0, 0, 0, -1308, -9088]$$ $$82881856/36015$$ $$107540213760$$ $$$$ $$12288$$ $$0.81289$$
10080.bc1 10080s3 $$[0, 0, 0, -10083, 389702]$$ $$303735479048/105$$ $$39191040$$ $$$$ $$12288$$ $$0.81289$$
10080.bc4 10080s4 $$[0, 0, 0, -3, 17498]$$ $$-8/354375$$ $$-132269760000$$ $$$$ $$12288$$ $$0.81289$$

## Rank

sage: E.rank()

The elliptic curves in class 10080s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10080s do not have complex multiplication.

## Modular form 10080.2.a.s

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} + 4q^{11} + 6q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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. 