# Properties

 Label 47040bq Number of curves $6$ Conductor $47040$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("47040.cd1")

sage: E.isogeny_class()

## Elliptic curves in class 47040bq

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.cd6 47040bq1 [0, -1, 0, 31295, 2971585] [2] 294912 $$\Gamma_0(N)$$-optimal
47040.cd5 47040bq2 [0, -1, 0, -219585, 30919617] [2, 2] 589824
47040.cd4 47040bq3 [0, -1, 0, -1160385, -454345023] [2] 1179648
47040.cd2 47040bq4 [0, -1, 0, -3292865, 2300844225] [2, 2] 1179648
47040.cd3 47040bq5 [0, -1, 0, -3073345, 2620597057] [2] 2359296
47040.cd1 47040bq6 [0, -1, 0, -52684865, 147207093825] [4] 2359296

## Rank

sage: E.rank()

The elliptic curves in class 47040bq have rank $$1$$.

## Modular form 47040.2.a.cd

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 4q^{11} - 2q^{13} - q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.