# Properties

 Label 75712.bx Number of curves $4$ Conductor $75712$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bx1")

sage: E.isogeny_class()

## Elliptic curves in class 75712.bx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.bx1 75712y4 $$[0, 0, 0, -639589964, 6223771098480]$$ $$22868021811807457713/8953460393696$$ $$11328983717494232465801216$$ $$[2]$$ $$23224320$$ $$3.7722$$
75712.bx2 75712y3 $$[0, 0, 0, -338472524, -2350626226832]$$ $$3389174547561866673/74853681183008$$ $$94713786405296186712915968$$ $$[2]$$ $$23224320$$ $$3.7722$$
75712.bx3 75712y2 $$[0, 0, 0, -46007884, 65950600560]$$ $$8511781274893233/3440817243136$$ $$4353731496908956113043456$$ $$[2, 2]$$ $$11612160$$ $$3.4256$$
75712.bx4 75712y1 $$[0, 0, 0, 9370036, 7493668208]$$ $$71903073502287/60782804992$$ $$-76909639153911209132032$$ $$[2]$$ $$5806080$$ $$3.0791$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 75712.bx have rank $$0$$.

## Complex multiplication

The elliptic curves in class 75712.bx do not have complex multiplication.

## Modular form 75712.2.a.bx

sage: E.q_eigenform(10)

$$q + 2q^{5} + q^{7} - 3q^{9} + 4q^{11} - 6q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.