Properties

 Label 800.e Number of curves $2$ Conductor $800$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

Elliptic curves in class 800.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
800.e1 800h1 $$[0, 0, 0, -5, 0]$$ $$1728$$ $$8000$$ $$[2]$$ $$32$$ $$-0.56160$$ $$\Gamma_0(N)$$-optimal $$-4$$
800.e2 800h2 $$[0, 0, 0, 20, 0]$$ $$1728$$ $$-512000$$ $$[2]$$ $$64$$ $$-0.21503$$   $$-4$$

Rank

sage: E.rank()

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

Complex multiplication

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

Modular form800.2.a.e

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.