# Properties

 Label 41616ci Number of curves 6 Conductor 41616 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("41616.s1")

sage: E.isogeny_class()

## Elliptic curves in class 41616ci

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
41616.s5 41616ci1 [0, 0, 0, -1415811, 601361026] [2] 884736 $$\Gamma_0(N)$$-optimal
41616.s4 41616ci2 [0, 0, 0, -4745091, -3281245310] [2, 2] 1769472
41616.s6 41616ci3 [0, 0, 0, 9404349, -19108808894] [2] 3538944
41616.s2 41616ci4 [0, 0, 0, -72163011, -235940487230] [2, 2] 3538944
41616.s3 41616ci5 [0, 0, 0, -68417571, -261524089694] [2] 7077888
41616.s1 41616ci6 [0, 0, 0, -1154595171, -15100548367646] [2] 7077888

## Rank

sage: E.rank()

The elliptic curves in class 41616ci have rank $$1$$.

## Modular form 41616.2.a.s

sage: E.q_eigenform(10)

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

## 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.