# Properties

 Label 5040bo Number of curves $4$ Conductor $5040$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bo1")

sage: E.isogeny_class()

## Elliptic curves in class 5040bo

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.bg3 5040bo1 [0, 0, 0, -507, -2774]  3072 $$\Gamma_0(N)$$-optimal
5040.bg2 5040bo2 [0, 0, 0, -3387, 73834] [2, 2] 6144
5040.bg1 5040bo3 [0, 0, 0, -53787, 4801354]  12288
5040.bg4 5040bo4 [0, 0, 0, 933, 249226]  12288

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 5040bo do not have complex multiplication.

## Modular form5040.2.a.bo

sage: E.q_eigenform(10)

$$q + q^{5} + q^{7} - 4q^{11} - 2q^{13} + 6q^{17} + O(q^{20})$$

## 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. 