# Properties

 Label 105966.ce Number of curves $6$ Conductor $105966$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("105966.ce1")

sage: E.isogeny_class()

## Elliptic curves in class 105966.ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
105966.ce1 105966bu4 [1, -1, 1, -10172894, -12486088879] [2] 3211264
105966.ce2 105966bu6 [1, -1, 1, -6918224, 6938387201] [2] 6422528
105966.ce3 105966bu3 [1, -1, 1, -787334, -94969807] [2, 2] 3211264
105966.ce4 105966bu2 [1, -1, 1, -635954, -194880607] [2, 2] 1605632
105966.ce5 105966bu1 [1, -1, 1, -30434, -4505119] [2] 802816 $$\Gamma_0(N)$$-optimal
105966.ce6 105966bu5 [1, -1, 1, 2921476, -735852175] [2] 6422528

## Rank

sage: E.rank()

The elliptic curves in class 105966.ce have rank $$1$$.

## Modular form 105966.2.a.ce

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2q^{5} - q^{7} + q^{8} + 2q^{10} - 4q^{11} + 6q^{13} - q^{14} + q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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 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.