# Properties

 Label 800a Number of curves $4$ Conductor $800$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 800a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
800.d3 800a1 $$[0, 0, 0, -25, 0]$$ $$1728$$ $$1000000$$ $$[2, 2]$$ $$64$$ $$-0.15924$$ $$\Gamma_0(N)$$-optimal $$-4$$
800.d1 800a2 $$[0, 0, 0, -275, -1750]$$ $$287496$$ $$8000000$$ $$$$ $$128$$ $$0.18733$$   $$-16$$
800.d2 800a3 $$[0, 0, 0, -275, 1750]$$ $$287496$$ $$8000000$$ $$$$ $$128$$ $$0.18733$$   $$-16$$
800.d4 800a4 $$[0, 0, 0, 100, 0]$$ $$1728$$ $$-64000000$$ $$$$ $$128$$ $$0.18733$$   $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 800a have rank $$1$$.

## Complex multiplication

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

## Modular form800.2.a.a

sage: E.q_eigenform(10)

$$q - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 