# Properties

 Label 16170cf Number of curves 4 Conductor 16170 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("16170.cd1")

sage: E.isogeny_class()

## Elliptic curves in class 16170cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
16170.cd3 16170cf1 [1, 0, 0, -24550, 1283540] [4] 73728 $$\Gamma_0(N)$$-optimal
16170.cd2 16170cf2 [1, 0, 0, -103930, -11623648] [2, 2] 147456
16170.cd1 16170cf3 [1, 0, 0, -1616560, -791233150] [2] 294912
16170.cd4 16170cf4 [1, 0, 0, 138620, -57756658] [2] 294912

## Rank

sage: E.rank()

The elliptic curves in class 16170cf have rank $$1$$.

## Modular form 16170.2.a.cd

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2q^{13} + q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.