# Properties

 Label 121680.eg Number of curves $6$ Conductor $121680$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("121680.eg1")

sage: E.isogeny_class()

## Elliptic curves in class 121680.eg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121680.eg1 121680ep6 [0, 0, 0, -219389547, -1250754169574] [2] 16515072
121680.eg2 121680ep4 [0, 0, 0, -20564427, 35865765754] [4] 8257536
121680.eg3 121680ep3 [0, 0, 0, -13750347, -19427767814] [2, 2] 8257536
121680.eg4 121680ep5 [0, 0, 0, -2799147, -49523855654] [2] 16515072
121680.eg5 121680ep2 [0, 0, 0, -1582347, 281958586] [2, 2] 4128768
121680.eg6 121680ep1 [0, 0, 0, 364533, 33926074] [2] 2064384 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 121680.eg have rank $$0$$.

## Modular form 121680.2.a.eg

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.