# Properties

 Label 12882.e Number of curves $2$ Conductor $12882$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 12882.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12882.e1 12882e1 $$[1, 1, 0, -494883, -79253235]$$ $$13403946614821979039929/5057590268826067968$$ $$5057590268826067968$$ $$$$ $$632320$$ $$2.2885$$ $$\Gamma_0(N)$$-optimal
12882.e2 12882e2 $$[1, 1, 0, 1548157, -563453715]$$ $$410363075617640914325831/374944243169850027552$$ $$-374944243169850027552$$ $$$$ $$1264640$$ $$2.6351$$

## Rank

sage: E.rank()

The elliptic curves in class 12882.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 12882.e do not have complex multiplication.

## Modular form 12882.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} - 4q^{10} - q^{12} + 4q^{14} - 4q^{15} + q^{16} - 6q^{17} - q^{18} + q^{19} + O(q^{20})$$ ## 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. 