# Properties

 Label 333270.ea Number of curves $4$ Conductor $333270$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ea1")

sage: E.isogeny_class()

## Elliptic curves in class 333270.ea

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.ea1 333270ea3 $$[1, -1, 1, -598188803, 3157747962027]$$ $$219353215817909485369/87028564162480920$$ $$9391962779991888106110914520$$ $$[2]$$ $$311427072$$ $$4.0658$$
333270.ea2 333270ea2 $$[1, -1, 1, -271584203, -1687888525413]$$ $$20527812941011798969/474091398849600$$ $$51163072896350548591617600$$ $$[2, 2]$$ $$155713536$$ $$3.7192$$
333270.ea3 333270ea1 $$[1, -1, 1, -270060683, -1708137325029]$$ $$20184279492242626489/11148103680$$ $$1203082870982136238080$$ $$[2]$$ $$77856768$$ $$3.3727$$ $$\Gamma_0(N)$$-optimal
333270.ea4 333270ea4 $$[1, -1, 1, 30644077, -5237620119669]$$ $$29489595518609351/109830613939935000$$ $$-11852718106456257268539735000$$ $$[2]$$ $$311427072$$ $$4.0658$$

## Rank

sage: E.rank()

The elliptic curves in class 333270.ea have rank $$1$$.

## Complex multiplication

The elliptic curves in class 333270.ea do not have complex multiplication.

## Modular form 333270.2.a.ea

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + 4q^{11} + 6q^{13} + q^{14} + q^{16} + 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.