# Properties

 Label 27380d Number of curves 4 Conductor 27380 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27380.c1")

sage: E.isogeny_class()

## Elliptic curves in class 27380d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27380.c3 27380d1 [0, 1, 0, -1825, 20028]  25920 $$\Gamma_0(N)$$-optimal
27380.c4 27380d2 [0, 1, 0, 5020, 140500]  51840
27380.c1 27380d3 [0, 1, 0, -56585, -5198600]  77760
27380.c2 27380d4 [0, 1, 0, -49740, -6496412]  155520

## Rank

sage: E.rank()

The elliptic curves in class 27380d have rank $$1$$.

## Modular form 27380.2.a.c

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{5} + 2q^{7} + q^{9} - 2q^{13} - 2q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 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. 