# Properties

 Label 4056.a Number of curves $2$ Conductor $4056$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 4056.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4056.a1 4056d1 $$[0, -1, 0, -110075, 13242696]$$ $$1909913257984/129730653$$ $$10018961335620432$$ $$[2]$$ $$40320$$ $$1.8189$$ $$\Gamma_0(N)$$-optimal
4056.a2 4056d2 $$[0, -1, 0, 95260, 56773716]$$ $$77366117936/1172914587$$ $$-1449327279299298048$$ $$[2]$$ $$80640$$ $$2.1654$$

## Rank

sage: E.rank()

The elliptic curves in class 4056.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4056.a do not have complex multiplication.

## Modular form4056.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - 4 q^{5} + q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 8 q^{19} + O(q^{20})$$

## 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.