# Properties

 Label 5040y Number of curves $4$ Conductor $5040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5040.z1")

sage: E.isogeny_class()

## Elliptic curves in class 5040y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.z3 5040y1 [0, 0, 0, -747, -6886] [2] 2304 $$\Gamma_0(N)$$-optimal
5040.z4 5040y2 [0, 0, 0, 1173, -36454] [2] 4608
5040.z1 5040y3 [0, 0, 0, -15147, 716634] [2] 6912
5040.z2 5040y4 [0, 0, 0, -10827, 1133946] [2] 13824

## Rank

sage: E.rank()

The elliptic curves in class 5040y have rank $$1$$.

## Modular form5040.2.a.z

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.