# Properties

 Label 38115k Number of curves 4 Conductor 38115 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("38115.v1")

sage: E.isogeny_class()

## Elliptic curves in class 38115k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38115.v3 38115k1 [1, -1, 0, -2745, 53136]  46080 $$\Gamma_0(N)$$-optimal
38115.v2 38115k2 [1, -1, 0, -8190, -218025] [2, 2] 92160
38115.v4 38115k3 [1, -1, 0, 19035, -1377810]  184320
38115.v1 38115k4 [1, -1, 0, -122535, -16477884]  184320

## Rank

sage: E.rank()

The elliptic curves in class 38115k have rank $$1$$.

## Modular form 38115.2.a.v

sage: E.q_eigenform(10)

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

## 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. 