# Properties

 Label 40931.a Number of curves 3 Conductor 40931 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("40931.a1")

sage: E.isogeny_class()

## Elliptic curves in class 40931.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40931.a1 40931a3 [0, -1, 1, -29099460, -60409576035] [] 1110000
40931.a2 40931a2 [0, -1, 1, -38450, -5243475] [] 222000
40931.a3 40931a1 [0, -1, 1, -1240, 40345] [] 44400 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 40931.a have rank $$1$$.

## Modular form 40931.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + 2q^{7} - 2q^{9} + 2q^{10} - q^{11} - 2q^{12} + 4q^{13} + 4q^{14} - q^{15} - 4q^{16} + 2q^{17} - 4q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 