# Properties

 Label 576i Number of curves $6$ Conductor $576$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 576i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
576.b5 576i1 $$[0, 0, 0, 24, 56]$$ $$2048/3$$ $$-2239488$$ $$[2]$$ $$64$$ $$-0.096046$$ $$\Gamma_0(N)$$-optimal
576.b4 576i2 $$[0, 0, 0, -156, 560]$$ $$35152/9$$ $$107495424$$ $$[2, 2]$$ $$128$$ $$0.25053$$
576.b3 576i3 $$[0, 0, 0, -876, -9520]$$ $$1556068/81$$ $$3869835264$$ $$[2, 2]$$ $$256$$ $$0.59710$$
576.b2 576i4 $$[0, 0, 0, -2316, 42896]$$ $$28756228/3$$ $$143327232$$ $$[2]$$ $$256$$ $$0.59710$$
576.b1 576i5 $$[0, 0, 0, -13836, -626416]$$ $$3065617154/9$$ $$859963392$$ $$[2]$$ $$512$$ $$0.94368$$
576.b6 576i6 $$[0, 0, 0, 564, -37744]$$ $$207646/6561$$ $$-626913312768$$ $$[2]$$ $$512$$ $$0.94368$$

## Rank

sage: E.rank()

The elliptic curves in class 576i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 576i do not have complex multiplication.

## Modular form576.2.a.i

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.