# Properties

 Label 1008i Number of curves 6 Conductor 1008 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("1008.h1")

sage: E.isogeny_class()

## Elliptic curves in class 1008i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1008.h5 1008i1 [0, 0, 0, -75, 506] [2] 192 $$\Gamma_0(N)$$-optimal
1008.h4 1008i2 [0, 0, 0, -1515, 22682] [2] 384
1008.h6 1008i3 [0, 0, 0, 645, -10582] [2] 576
1008.h3 1008i4 [0, 0, 0, -5115, -115414] [2] 1152
1008.h2 1008i5 [0, 0, 0, -24555, -1485286] [2] 1728
1008.h1 1008i6 [0, 0, 0, -393195, -94898662] [2] 3456

## Rank

sage: E.rank()

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

## Modular form1008.2.a.h

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.