# Properties

 Label 136710du Number of curves $6$ Conductor $136710$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("136710.dd1")

sage: E.isogeny_class()

## Elliptic curves in class 136710du

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
136710.dd6 136710du1 [1, -1, 0, 26451, -9387995] [2] 1572864 $$\Gamma_0(N)$$-optimal
136710.dd5 136710du2 [1, -1, 0, -538029, -143395547] [2, 2] 3145728
136710.dd4 136710du3 [1, -1, 0, -1631709, 625898965] [2] 6291456
136710.dd2 136710du4 [1, -1, 0, -8476029, -9495947147] [2, 2] 6291456
136710.dd3 136710du5 [1, -1, 0, -8343729, -9806825687] [2] 12582912
136710.dd1 136710du6 [1, -1, 0, -135616329, -607843627007] [2] 12582912

## Rank

sage: E.rank()

The elliptic curves in class 136710du have rank $$1$$.

## Modular form 136710.2.a.dd

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + 4q^{11} - 6q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.