# Properties

 Label 1008g Number of curves 4 Conductor 1008 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1008.e1")

sage: E.isogeny_class()

## Elliptic curves in class 1008g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1008.e3 1008g1 [0, 0, 0, -66, -205]  128 $$\Gamma_0(N)$$-optimal
1008.e2 1008g2 [0, 0, 0, -111, 110] [2, 2] 256
1008.e1 1008g3 [0, 0, 0, -1371, 19514]  512
1008.e4 1008g4 [0, 0, 0, 429, 866]  512

## Rank

sage: E.rank()

The elliptic curves in class 1008g have rank $$1$$.

## Modular form1008.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{5} + q^{7} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 