# Properties

 Label 16575.i Number of curves $6$ Conductor $16575$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("16575.i1")

sage: E.isogeny_class()

## Elliptic curves in class 16575.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
16575.i1 16575f5 [1, 0, 1, -504351, -137883527] [2] 131072
16575.i2 16575f3 [1, 0, 1, -34726, -1692277] [2, 2] 65536
16575.i3 16575f2 [1, 0, 1, -13601, 589223] [2, 2] 32768
16575.i4 16575f1 [1, 0, 1, -13476, 600973] [2] 16384 $$\Gamma_0(N)$$-optimal
16575.i5 16575f4 [1, 0, 1, 5524, 2119223] [2] 65536
16575.i6 16575f6 [1, 0, 1, 96899, -11432527] [2] 131072

## Rank

sage: E.rank()

The elliptic curves in class 16575.i have rank $$0$$.

## Modular form 16575.2.a.i

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + 4q^{11} - q^{12} - q^{13} - q^{16} - q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.