# Properties

 Label 229320ds Number of curves $4$ Conductor $229320$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ds1")

sage: E.isogeny_class()

## Elliptic curves in class 229320ds

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
229320.r4 229320ds1 $$[0, 0, 0, 19257, -12274598]$$ $$35969456/2985255$$ $$-65544637835738880$$ $$[2]$$ $$1966080$$ $$1.9063$$ $$\Gamma_0(N)$$-optimal
229320.r3 229320ds2 $$[0, 0, 0, -695163, -215312762]$$ $$423026849956/16769025$$ $$1472731368654873600$$ $$[2, 2]$$ $$3932160$$ $$2.2529$$
229320.r2 229320ds3 $$[0, 0, 0, -1806483, 645071182]$$ $$3711757787138/1124589375$$ $$197533016906883840000$$ $$[2]$$ $$7864320$$ $$2.5994$$
229320.r1 229320ds4 $$[0, 0, 0, -11014563, -14070139202]$$ $$841356017734178/1404585$$ $$246713972868679680$$ $$[2]$$ $$7864320$$ $$2.5994$$

## Rank

sage: E.rank()

The elliptic curves in class 229320ds have rank $$0$$.

## Complex multiplication

The elliptic curves in class 229320ds do not have complex multiplication.

## Modular form 229320.2.a.ds

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.