# Properties

 Label 8400.w Number of curves $2$ Conductor $8400$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 8400.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8400.w1 8400bv1 $$[0, -1, 0, -32208, -257088]$$ $$461889917/263424$$ $$2107392000000000$$ $$$$ $$46080$$ $$1.6301$$ $$\Gamma_0(N)$$-optimal
8400.w2 8400bv2 $$[0, -1, 0, 127792, -2177088]$$ $$28849701763/16941456$$ $$-135531648000000000$$ $$$$ $$92160$$ $$1.9767$$

## Rank

sage: E.rank()

The elliptic curves in class 8400.w have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8400.w do not have complex multiplication.

## Modular form8400.2.a.w

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} - 2q^{11} + 6q^{13} + 4q^{17} + 6q^{19} + O(q^{20})$$ ## 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. 