# Properties

 Label 106575.cg Number of curves $6$ Conductor $106575$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("106575.cg1")

sage: E.isogeny_class()

## Elliptic curves in class 106575.cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106575.cg1 106575r4 [1, 1, 0, -250664425, -1527625367750] [2] 9437184
106575.cg2 106575r6 [1, 1, 0, -52024550, 117389126625] [2] 18874368
106575.cg3 106575r3 [1, 1, 0, -15966675, -22912065000] [2, 2] 9437184
106575.cg4 106575r2 [1, 1, 0, -15666550, -23873965625] [2, 2] 4718592
106575.cg5 106575r1 [1, 1, 0, -960425, -388284000] [2] 2359296 $$\Gamma_0(N)$$-optimal
106575.cg6 106575r5 [1, 1, 0, 15289200, -101645614125] [2] 18874368

## Rank

sage: E.rank()

The elliptic curves in class 106575.cg have rank $$1$$.

## Modular form 106575.2.a.cg

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.