# Properties

 Label 32175.j Number of curves $6$ Conductor $32175$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("32175.j1")

sage: E.isogeny_class()

## Elliptic curves in class 32175.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32175.j1 32175p4 [1, -1, 1, -1544405, -738350778] [2] 262144
32175.j2 32175p6 [1, -1, 1, -677030, 207796722] [2] 524288
32175.j3 32175p3 [1, -1, 1, -106655, -8945778] [2, 2] 262144
32175.j4 32175p2 [1, -1, 1, -96530, -11517528] [2, 2] 131072
32175.j5 32175p1 [1, -1, 1, -5405, -218028] [2] 65536 $$\Gamma_0(N)$$-optimal
32175.j6 32175p5 [1, -1, 1, 301720, -61217778] [2] 524288

## Rank

sage: E.rank()

The elliptic curves in class 32175.j have rank $$1$$.

## Modular form 32175.2.a.j

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.