# Properties

 Label 5040.g Number of curves 8 Conductor 5040 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("5040.g1")

sage: E.isogeny_class()

## Elliptic curves in class 5040.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.g1 5040bc7 [0, 0, 0, -50577483, -138447122182] [2] 221184
5040.g2 5040bc6 [0, 0, 0, -3161163, -2163135238] [2, 2] 110592
5040.g3 5040bc8 [0, 0, 0, -2930763, -2491823878] [2] 221184
5040.g4 5040bc4 [0, 0, 0, -627483, -187952182] [2] 73728
5040.g5 5040bc3 [0, 0, 0, -212043, -28562182] [2] 55296
5040.g6 5040bc2 [0, 0, 0, -83163, 4845962] [2, 2] 36864
5040.g7 5040bc1 [0, 0, 0, -71643, 7378058] [2] 18432 $$\Gamma_0(N)$$-optimal
5040.g8 5040bc5 [0, 0, 0, 276837, 35589962] [2] 73728

## Rank

sage: E.rank()

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

## Modular form5040.2.a.g

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.