# Properties

 Label 128.d Number of curves $2$ Conductor $128$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 128.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
128.d1 128d2 $$[0, -1, 0, -2, 2]$$ $$10976$$ $$128$$ $$[2]$$ $$16$$ $$-0.85643$$
128.d2 128d1 $$[0, -1, 0, 3, 5]$$ $$128$$ $$-16384$$ $$[2]$$ $$8$$ $$-0.50986$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 128.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 128.d do not have complex multiplication.

## Modular form128.2.a.d

sage: E.q_eigenform(10)

$$q + 2 q^{3} + 2 q^{5} - 4 q^{7} + q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} + 2 q^{19} + O(q^{20})$$

## 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.