# Properties

 Label 381150gy Number of curves $4$ Conductor $381150$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("381150.gy1")

sage: E.isogeny_class()

## Elliptic curves in class 381150gy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
381150.gy2 381150gy1 [1, -1, 0, -318192, -68919284]  3317760 $$\Gamma_0(N)$$-optimal
381150.gy3 381150gy2 [1, -1, 0, -227442, -109121534]  6635520
381150.gy1 381150gy3 [1, -1, 0, -1271067, 483642341]  9953280
381150.gy4 381150gy4 [1, -1, 0, 1995933, 2558187341]  19906560

## Rank

sage: E.rank()

The elliptic curves in class 381150gy have rank $$1$$.

## Modular form 381150.2.a.gy

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{7} - q^{8} + 2q^{13} - q^{14} + q^{16} - 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 