# Properties

 Label 270480.gs Number of curves $4$ Conductor $270480$ CM no Rank $0$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("gs1")

sage: E.isogeny_class()

## Elliptic curves in class 270480.gs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
270480.gs1 270480gs4 $$[0, 1, 0, -9088536, -10549060236]$$ $$344577854816148242/2716875$$ $$654617859840000$$ $$[2]$$ $$5898240$$ $$2.4339$$
270480.gs2 270480gs2 $$[0, 1, 0, -568416, -164737980]$$ $$168591300897604/472410225$$ $$56912476734489600$$ $$[2, 2]$$ $$2949120$$ $$2.0873$$
270480.gs3 270480gs3 $$[0, 1, 0, -343016, -296461740]$$ $$-18524646126002/146738831715$$ $$-35356010111873095680$$ $$[2]$$ $$5898240$$ $$2.4339$$
270480.gs4 270480gs1 $$[0, 1, 0, -49996, -295156]$$ $$458891455696/264449745$$ $$7964735500673280$$ $$[2]$$ $$1474560$$ $$1.7407$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 270480.gs have rank $$0$$.

## Complex multiplication

The elliptic curves in class 270480.gs do not have complex multiplication.

## Modular form 270480.2.a.gs

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.