# Properties

 Label 12675.n Number of curves 8 Conductor 12675 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12675.n1")

sage: E.isogeny_class()

## Elliptic curves in class 12675.n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12675.n1 12675x7 [1, 0, 0, -9126088, -10612219333] [2] 221184
12675.n2 12675x5 [1, 0, 0, -570463, -165801208] [2, 2] 110592
12675.n3 12675x8 [1, 0, 0, -464838, -229070583] [2] 221184
12675.n4 12675x4 [1, 0, 0, -338088, 75636417] [2] 55296
12675.n5 12675x3 [1, 0, 0, -42338, -1554333] [2, 2] 55296
12675.n6 12675x2 [1, 0, 0, -21213, 1170792] [2, 2] 27648
12675.n7 12675x1 [1, 0, 0, -88, 51167] [2] 13824 $$\Gamma_0(N)$$-optimal
12675.n8 12675x6 [1, 0, 0, 147787, -11630958] [2] 110592

## Rank

sage: E.rank()

The elliptic curves in class 12675.n have rank $$0$$.

## Modular form 12675.2.a.n

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.