# Properties

 Label 5040.i Number of curves $6$ Conductor $5040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5040.i1")

sage: E.isogeny_class()

## Elliptic curves in class 5040.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.i1 5040bg5 [0, 0, 0, -2419203, -1448293502] [2] 49152
5040.i2 5040bg3 [0, 0, 0, -151203, -22628702] [2, 2] 24576
5040.i3 5040bg6 [0, 0, 0, -141123, -25775678] [2] 49152
5040.i4 5040bg4 [0, 0, 0, -53283, 4474402] [2] 24576
5040.i5 5040bg2 [0, 0, 0, -10083, -303518] [2, 2] 12288
5040.i6 5040bg1 [0, 0, 0, 1437, -29342] [2] 6144 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form5040.2.a.i

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} + 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.