# Properties

 Label 38.a Number of curves $3$ Conductor $38$ CM no Rank $0$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 38.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38.a1 38a2 $$[1, 0, 1, -86, -2456]$$ $$-69173457625/2550136832$$ $$-2550136832$$ $$[]$$ $$18$$ $$0.48517$$
38.a2 38a3 $$[1, 0, 1, -16, 22]$$ $$-413493625/152$$ $$-152$$ $$$$ $$18$$ $$-0.61344$$
38.a3 38a1 $$[1, 0, 1, 9, 90]$$ $$94196375/3511808$$ $$-3511808$$ $$$$ $$6$$ $$-0.064137$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 38.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 38.a do not have complex multiplication.

## Modular form38.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2 q^{9} - 6 q^{11} + q^{12} + 5 q^{13} + q^{14} + q^{16} + 3 q^{17} + 2 q^{18} + q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 