# Properties

 Label 1859.b Number of curves $3$ Conductor $1859$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1859.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1859.b1 1859a3 $$[0, -1, 1, -1321636, -584371175]$$ $$-52893159101157376/11$$ $$-53094899$$ $$[]$$ $$10800$$ $$1.7792$$
1859.b2 1859a2 $$[0, -1, 1, -1746, -50295]$$ $$-122023936/161051$$ $$-777362416259$$ $$[]$$ $$2160$$ $$0.97446$$
1859.b3 1859a1 $$[0, -1, 1, -56, 405]$$ $$-4096/11$$ $$-53094899$$ $$[]$$ $$432$$ $$0.16975$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1859.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1859.b do not have complex multiplication.

## Modular form1859.2.a.b

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 