# Properties

 Label 117117r Number of curves 6 Conductor 117117 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("117117.l1")

sage: E.isogeny_class()

## Elliptic curves in class 117117r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
117117.l4 117117r1 [1, -1, 1, -51746, -4516248] [2] 307200 $$\Gamma_0(N)$$-optimal
117117.l3 117117r2 [1, -1, 1, -59351, -3095634] [2, 2] 614400
117117.l6 117117r3 [1, -1, 1, 191614, -22570518] [2] 1228800
117117.l2 117117r4 [1, -1, 1, -431996, 107207286] [2, 2] 1228800
117117.l5 117117r5 [1, -1, 1, 47119, 331624752] [2] 2457600
117117.l1 117117r6 [1, -1, 1, -6873431, 6937704960] [2] 2457600

## Rank

sage: E.rank()

The elliptic curves in class 117117r have rank $$2$$.

## Modular form 117117.2.a.l

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - 2q^{5} - q^{7} + 3q^{8} + 2q^{10} - q^{11} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.