# Properties

 Label 206310bg Number of curves $6$ Conductor $206310$ CM no Rank $0$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("206310.bi1")

sage: E.isogeny_class()

## Elliptic curves in class 206310bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
206310.bi6 206310bg1 [1, 1, 1, 7924, -105427] [2] 720896 $$\Gamma_0(N)$$-optimal
206310.bi5 206310bg2 [1, 1, 1, -34396, -917971] [2, 2] 1441792
206310.bi3 206310bg3 [1, 1, 1, -298896, 62138829] [2, 2] 2883584
206310.bi2 206310bg4 [1, 1, 1, -447016, -115131187] [2] 2883584
206310.bi1 206310bg5 [1, 1, 1, -4768946, 4006510949] [2] 5767168
206310.bi4 206310bg6 [1, 1, 1, -60846, 158691909] [2] 5767168

## Rank

sage: E.rank()

The elliptic curves in class 206310bg have rank $$0$$.

## Modular form 206310.2.a.bi

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} + q^{13} + q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.