# Properties

 Label 272734ce Number of curves $2$ Conductor $272734$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ce1")

sage: E.isogeny_class()

## Elliptic curves in class 272734ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
272734.ce2 272734ce1 $$[1, 0, 0, 204427, 167364385]$$ $$4533086375/60669952$$ $$-12644975795725385728$$ $$$$ $$6881280$$ $$2.3461$$ $$\Gamma_0(N)$$-optimal
272734.ce1 272734ce2 $$[1, 0, 0, -3590133, 2450930593]$$ $$24553362849625/1755162752$$ $$365815198215399244928$$ $$$$ $$13762560$$ $$2.6927$$

## Rank

sage: E.rank()

The elliptic curves in class 272734ce have rank $$2$$.

## Complex multiplication

The elliptic curves in class 272734ce do not have complex multiplication.

## Modular form 272734.2.a.ce

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + q^{8} + q^{9} - 2q^{12} + q^{16} + 6q^{17} + q^{18} - 6q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 