# Properties

 Label 38115.k Number of curves $6$ Conductor $38115$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("38115.k1")

sage: E.isogeny_class()

## Elliptic curves in class 38115.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38115.k1 38115s6 [1, -1, 1, -14429273, -21092093778] [2] 1966080
38115.k2 38115s4 [1, -1, 1, -952898, -289961328] [2, 2] 983040
38115.k3 38115s2 [1, -1, 1, -294053, 57381756] [2, 2] 491520
38115.k4 38115s1 [1, -1, 1, -288608, 59749242] [2] 245760 $$\Gamma_0(N)$$-optimal
38115.k5 38115s3 [1, -1, 1, 277672, 253140396] [2] 983040
38115.k6 38115s5 [1, -1, 1, 1981957, -1725692394] [2] 1966080

## Rank

sage: E.rank()

The elliptic curves in class 38115.k have rank $$0$$.

## Modular form 38115.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - q^{5} + q^{7} + 3q^{8} + q^{10} + 2q^{13} - q^{14} - q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.