Properties

 Label 6400.x Number of curves $2$ Conductor $6400$ CM $$\Q(\sqrt{-2})$$ Rank $1$ Graph

Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

Elliptic curves in class 6400.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
6400.x1 6400f2 $$[0, -1, 0, -333, -1963]$$ $$8000$$ $$512000000$$ $$[2]$$ $$2304$$ $$0.40075$$   $$-8$$
6400.x2 6400f1 $$[0, -1, 0, -83, 287]$$ $$8000$$ $$8000000$$ $$[2]$$ $$1152$$ $$0.054173$$ $$\Gamma_0(N)$$-optimal $$-8$$

Rank

sage: E.rank()

The elliptic curves in class 6400.x have rank $$1$$.

Complex multiplication

Each elliptic curve in class 6400.x has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-2})$$.

Modular form6400.2.a.x

sage: E.q_eigenform(10)

$$q + 2q^{3} + q^{9} - 6q^{11} + 6q^{17} - 2q^{19} + O(q^{20})$$

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.