# Properties

 Label 800.f Number of curves $2$ Conductor $800$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 800.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
800.f1 800d1 $$[0, 0, 0, -125, 0]$$ $$1728$$ $$125000000$$ $$$$ $$160$$ $$0.24312$$ $$\Gamma_0(N)$$-optimal $$-4$$
800.f2 800d2 $$[0, 0, 0, 500, 0]$$ $$1728$$ $$-8000000000$$ $$$$ $$320$$ $$0.58969$$   $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 800.f have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 800.f has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form800.2.a.f

sage: E.q_eigenform(10)

$$q - 3q^{9} + 4q^{13} + 8q^{17} + O(q^{20})$$ ## 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 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. 