# Properties

 Label 5040bj Number of curves $8$ Conductor $5040$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5040.ba1")

sage: E.isogeny_class()

## Elliptic curves in class 5040bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.ba7 5040bj1 [0, 0, 0, -5907, -61486] [2] 9216 $$\Gamma_0(N)$$-optimal
5040.ba5 5040bj2 [0, 0, 0, -51987, 4518866] [2, 2] 18432
5040.ba4 5040bj3 [0, 0, 0, -386067, -92329774] [2] 27648
5040.ba2 5040bj4 [0, 0, 0, -829587, 290831186] [2] 36864
5040.ba6 5040bj5 [0, 0, 0, -11667, 11349074] [2] 36864
5040.ba3 5040bj6 [0, 0, 0, -388947, -90882286] [2, 2] 55296
5040.ba1 5040bj7 [0, 0, 0, -928947, 216809714] [2] 110592
5040.ba8 5040bj8 [0, 0, 0, 104973, -305935054] [2] 110592

## Rank

sage: E.rank()

The elliptic curves in class 5040bj have rank $$0$$.

## Modular form5040.2.a.ba

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} + 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.