# Properties

 Label 422331.bj Number of curves $2$ Conductor $422331$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bj1")

sage: E.isogeny_class()

## Elliptic curves in class 422331.bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422331.bj1 422331bj1 $$[1, 1, 0, -3768027, -2815342200]$$ $$10418796526321/6390657$$ $$3629057610640580937$$ $$$$ $$18063360$$ $$2.5041$$ $$\Gamma_0(N)$$-optimal
422331.bj2 422331bj2 $$[1, 1, 0, -3064142, -3898621215]$$ $$-5602762882081/8312741073$$ $$-4720550055535009779993$$ $$$$ $$36126720$$ $$2.8507$$

## Rank

sage: E.rank()

The elliptic curves in class 422331.bj have rank $$0$$.

## Complex multiplication

The elliptic curves in class 422331.bj do not have complex multiplication.

## Modular form 422331.2.a.bj

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 