# Properties

 Label 53040.s Number of curves 8 Conductor 53040 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("53040.s1")

sage: E.isogeny_class()

## Elliptic curves in class 53040.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53040.s1 53040bp8 [0, -1, 0, -105456000096, 13181238367312896] [4] 79626240
53040.s2 53040bp6 [0, -1, 0, -6591000096, 205958491312896] [2, 2] 39813120
53040.s3 53040bp7 [0, -1, 0, -6579048416, 206742626695680] [2] 79626240
53040.s4 53040bp5 [0, -1, 0, -1301972256, 18080294342400] [4] 26542080
53040.s5 53040bp3 [0, -1, 0, -412684576, 3205946556160] [2] 19906560
53040.s6 53040bp2 [0, -1, 0, -88916256, 227021356800] [2, 2] 13271040
53040.s7 53040bp1 [0, -1, 0, -33538336, -71975108864] [2] 6635520 $$\Gamma_0(N)$$-optimal
53040.s8 53040bp4 [0, -1, 0, 238093024, 1508636126976] [2] 26542080

## Rank

sage: E.rank()

The elliptic curves in class 53040.s have rank $$0$$.

## Modular form 53040.2.a.s

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + 4q^{7} + q^{9} + q^{13} + q^{15} + q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.