# Properties

 Label 5082.s Number of curves $2$ Conductor $5082$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5082.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5082.s1 5082t2 $$[1, 1, 1, -12768, 523227]$$ $$129938649625/7072758$$ $$12529822235238$$ $$[2]$$ $$15360$$ $$1.2692$$
5082.s2 5082t1 $$[1, 1, 1, 542, 33419]$$ $$9938375/274428$$ $$-486165942108$$ $$[2]$$ $$7680$$ $$0.92258$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5082.s have rank $$0$$.

## Complex multiplication

The elliptic curves in class 5082.s do not have complex multiplication.

## Modular form5082.2.a.s

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + 2 q^{13} + q^{14} + q^{16} + 4 q^{17} + q^{18} - 6 q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels.