# Properties

 Label 38b Number of curves $2$ Conductor $38$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 38b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38.b2 38b1 $$[1, 1, 1, 0, 1]$$ $$-1/608$$ $$-608$$ $$[5]$$ $$2$$ $$-0.78693$$ $$\Gamma_0(N)$$-optimal
38.b1 38b2 $$[1, 1, 1, -70, -279]$$ $$-37966934881/4952198$$ $$-4952198$$ $$[]$$ $$10$$ $$0.017785$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form38.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} - 4 q^{10} + 2 q^{11} - q^{12} - q^{13} + 3 q^{14} + 4 q^{15} + q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.