# Properties

 Label 116928dw Number of curves $6$ Conductor $116928$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("116928.r1")

sage: E.isogeny_class()

## Elliptic curves in class 116928dw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
116928.r5 116928dw1 [0, 0, 0, -451596, 125063696] [2] 1572864 $$\Gamma_0(N)$$-optimal
116928.r4 116928dw2 [0, 0, 0, -7366476, 7695474320] [2, 2] 3145728
116928.r3 116928dw3 [0, 0, 0, -7507596, 7385292560] [2, 2] 6291456
116928.r1 116928dw4 [0, 0, 0, -117863436, 492511936016] [2] 6291456
116928.r6 116928dw5 [0, 0, 0, 7189044, 32775207824] [2] 12582912
116928.r2 116928dw6 [0, 0, 0, -24462156, -37856255344] [2] 12582912

## Rank

sage: E.rank()

The elliptic curves in class 116928dw have rank $$0$$.

## Modular form 116928.2.a.r

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.