# Properties

 Label 51842i Number of curves $2$ Conductor $51842$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 51842i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51842.d2 51842i1 $$[1, 0, 1, 893734, 1529837964]$$ $$4533086375/60669952$$ $$-1056644526100817231872$$ $$$$ $$2838528$$ $$2.7149$$ $$\Gamma_0(N)$$-optimal
51842.d1 51842i2 $$[1, 0, 1, -15695706, 22405989260]$$ $$24553362849625/1755162752$$ $$30568395938682236012672$$ $$$$ $$5677056$$ $$3.0615$$

## Rank

sage: E.rank()

The elliptic curves in class 51842i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 51842i do not have complex multiplication.

## Modular form 51842.2.a.i

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} - q^{8} + q^{9} - 4q^{11} - 2q^{12} + q^{16} + 6q^{17} - q^{18} - 6q^{19} + O(q^{20})$$ ## 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. 