# Properties

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

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 390.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
390.a1 390a3 $$[1, 1, 0, -483, -4293]$$ $$12501706118329/2570490$$ $$2570490$$ $$$$ $$128$$ $$0.22703$$
390.a2 390a2 $$[1, 1, 0, -33, -63]$$ $$4165509529/1368900$$ $$1368900$$ $$[2, 2]$$ $$64$$ $$-0.11954$$
390.a3 390a1 $$[1, 1, 0, -13, 13]$$ $$273359449/9360$$ $$9360$$ $$$$ $$32$$ $$-0.46611$$ $$\Gamma_0(N)$$-optimal
390.a4 390a4 $$[1, 1, 0, 97, -297]$$ $$99317171591/106616250$$ $$-106616250$$ $$$$ $$128$$ $$0.22703$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 390.a do not have complex multiplication.

## Modular form390.2.a.a

sage: E.q_eigenform(10)

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

## 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. 