# Properties

 Label 333270.en Number of curves $2$ Conductor $333270$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("en1")

sage: E.isogeny_class()

## Elliptic curves in class 333270.en

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.en1 333270en2 $$[1, -1, 1, -2011622, 612666119]$$ $$8341959848041/3327411150$$ $$359088099123237753150$$ $$$$ $$16220160$$ $$2.6424$$
333270.en2 333270en1 $$[1, -1, 1, -916592, -330811729]$$ $$789145184521/17996580$$ $$1942157855340262980$$ $$$$ $$8110080$$ $$2.2958$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 333270.2.a.en

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + 2q^{11} - 2q^{13} - q^{14} + q^{16} + 6q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 