# Properties

 Label 47040.db Number of curves $8$ Conductor $47040$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.db1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.db

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.db1 47040ey8 [0, -1, 0, -1101465185, -14069962595583] [2] 10616832
47040.db2 47040ey6 [0, -1, 0, -68843105, -219815685375] [2, 2] 5308416
47040.db3 47040ey7 [0, -1, 0, -63825505, -253221862655] [2] 10616832
47040.db4 47040ey5 [0, -1, 0, -13665185, -19096955583] [2] 3538944
47040.db5 47040ey3 [0, -1, 0, -4617825, -2901224703] [2] 2654208
47040.db6 47040ey2 [0, -1, 0, -1811105, 493097025] [2, 2] 1769472
47040.db7 47040ey1 [0, -1, 0, -1560225, 750349377] [2] 884736 $$\Gamma_0(N)$$-optimal
47040.db8 47040ey4 [0, -1, 0, 6028895, 3614985025] [2] 3538944

## Rank

sage: E.rank()

The elliptic curves in class 47040.db have rank $$0$$.

## Modular form 47040.2.a.db

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} + 2q^{13} - q^{15} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.