# Properties

 Label 221760be Number of curves $6$ Conductor $221760$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("221760.mb1")

sage: E.isogeny_class()

## Elliptic curves in class 221760be

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221760.mb4 221760be1 [0, 0, 0, -152652, 22956176] [2] 1048576 $$\Gamma_0(N)$$-optimal
221760.mb3 221760be2 [0, 0, 0, -155532, 22044944] [2, 2] 2097152
221760.mb5 221760be3 [0, 0, 0, 146868, 97403024] [2] 4194304
221760.mb2 221760be4 [0, 0, 0, -504012, -111631984] [2, 2] 4194304
221760.mb6 221760be5 [0, 0, 0, 1048308, -663636976] [2] 8388608
221760.mb1 221760be6 [0, 0, 0, -7632012, -8114950384] [2] 8388608

## Rank

sage: E.rank()

The elliptic curves in class 221760be have rank $$1$$.

## Modular form 221760.2.a.mb

sage: E.q_eigenform(10)

$$q + q^{5} + q^{7} - q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.