# Properties

 Label 27720p Number of curves $4$ Conductor $27720$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 27720p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27720.y4 27720p1 $$[0, 0, 0, 3273, -69046]$$ $$20777545136/23059575$$ $$-4303470124800$$ $$[2]$$ $$32768$$ $$1.1107$$ $$\Gamma_0(N)$$-optimal
27720.y3 27720p2 $$[0, 0, 0, -18507, -648394]$$ $$939083699236/300155625$$ $$224064973440000$$ $$[2, 2]$$ $$65536$$ $$1.4573$$
27720.y2 27720p3 $$[0, 0, 0, -117507, 15013406]$$ $$120186986927618/4332064275$$ $$6467737306060800$$ $$[2]$$ $$131072$$ $$1.8039$$
27720.y1 27720p4 $$[0, 0, 0, -267987, -53388466]$$ $$1425631925916578/270703125$$ $$404157600000000$$ $$[2]$$ $$131072$$ $$1.8039$$

## Rank

sage: E.rank()

The elliptic curves in class 27720p have rank $$1$$.

## Complex multiplication

The elliptic curves in class 27720p do not have complex multiplication.

## Modular form 27720.2.a.p

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} - q^{11} - 2 q^{13} + 2 q^{17} + O(q^{20})$$

## 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 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.