# Properties

 Label 123210.dg Number of curves $6$ Conductor $123210$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("123210.dg1")

sage: E.isogeny_class()

## Elliptic curves in class 123210.dg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123210.dg1 123210dh4 [1, -1, 1, -4201522862, -104822341364739] [2] 75644928
123210.dg2 123210dh6 [1, -1, 1, -3734803382, 87470606300829] [2] 151289856
123210.dg3 123210dh3 [1, -1, 1, -361313582, -296803127811] [2, 2] 75644928
123210.dg4 123210dh2 [1, -1, 1, -262745582, -1635829694211] [2, 2] 37822464
123210.dg5 123210dh1 [1, -1, 1, -10411502, -44510052099] [4] 18911232 $$\Gamma_0(N)$$-optimal
123210.dg6 123210dh5 [1, -1, 1, 1435088218, -2367695122851] [2] 151289856

## Rank

sage: E.rank()

The elliptic curves in class 123210.dg have rank $$0$$.

## Modular form 123210.2.a.dg

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.