# Properties

 Label 53040cz Number of curves 4 Conductor 53040 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("53040.cj1")

sage: E.isogeny_class()

## Elliptic curves in class 53040cz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53040.cj4 53040cz1 [0, 1, 0, 25280, -253900] [2] 276480 $$\Gamma_0(N)$$-optimal
53040.cj3 53040cz2 [0, 1, 0, -102720, -2148300] [2, 2] 552960
53040.cj2 53040cz3 [0, 1, 0, -1027520, 398475060] [4] 1105920
53040.cj1 53040cz4 [0, 1, 0, -1225920, -521965260] [2] 1105920

## Rank

sage: E.rank()

The elliptic curves in class 53040cz have rank $$1$$.

## Modular form 53040.2.a.cj

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - 4q^{7} + q^{9} + 4q^{11} + q^{13} + q^{15} - q^{17} - 8q^{19} + O(q^{20})$$

## 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.