# Properties

 Label 47040gk Number of curves $4$ Conductor $47040$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gk1")

sage: E.isogeny_class()

## Elliptic curves in class 47040gk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.dz3 47040gk1 [0, 1, 0, -11041, -285601]  147456 $$\Gamma_0(N)$$-optimal
47040.dz2 47040gk2 [0, 1, 0, -73761, 7479135] [2, 2] 294912
47040.dz4 47040gk3 [0, 1, 0, 20319, 25335519]  589824
47040.dz1 47040gk4 [0, 1, 0, -1171361, 487569375]  589824

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 47040gk do not have complex multiplication.

## Modular form 47040.2.a.gk

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + q^{9} - 4q^{11} - 2q^{13} - q^{15} + 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 