# Properties

 Label 390.a Number of curves $4$ Conductor $390$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("390.a1")

sage: E.isogeny_class()

## Elliptic curves in class 390.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
390.a1 390a3 [1, 1, 0, -483, -4293]  128
390.a2 390a2 [1, 1, 0, -33, -63] [2, 2] 64
390.a3 390a1 [1, 1, 0, -13, 13]  32 $$\Gamma_0(N)$$-optimal
390.a4 390a4 [1, 1, 0, 97, -297]  128

## Rank

sage: E.rank()

The elliptic curves in class 390.a have rank $$1$$.

## Modular form390.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{12} - q^{13} + q^{15} + q^{16} - 6q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 