# Properties

 Label 23232.do Number of curves $6$ Conductor $23232$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("do1")

sage: E.isogeny_class()

## Elliptic curves in class 23232.do

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
23232.do1 23232bw6 $$[0, 1, 0, -186017, 30817983]$$ $$3065617154/9$$ $$2089818390528$$ $$[2]$$ $$81920$$ $$1.5933$$
23232.do2 23232bw4 $$[0, 1, 0, -31137, -2124993]$$ $$28756228/3$$ $$348303065088$$ $$[2]$$ $$40960$$ $$1.2467$$
23232.do3 23232bw3 $$[0, 1, 0, -11777, 465375]$$ $$1556068/81$$ $$9404182757376$$ $$[2, 2]$$ $$40960$$ $$1.2467$$
23232.do4 23232bw2 $$[0, 1, 0, -2097, -28305]$$ $$35152/9$$ $$261227298816$$ $$[2, 2]$$ $$20480$$ $$0.90017$$
23232.do5 23232bw1 $$[0, 1, 0, 323, -2653]$$ $$2048/3$$ $$-5442235392$$ $$[2]$$ $$10240$$ $$0.55360$$ $$\Gamma_0(N)$$-optimal
23232.do6 23232bw5 $$[0, 1, 0, 7583, 1863167]$$ $$207646/6561$$ $$-1523477606694912$$ $$[2]$$ $$81920$$ $$1.5933$$

## Rank

sage: E.rank()

The elliptic curves in class 23232.do have rank $$1$$.

## Complex multiplication

The elliptic curves in class 23232.do do not have complex multiplication.

## Modular form 23232.2.a.do

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} - 2q^{13} + 2q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.