# Properties

 Label 75712.u Number of curves $2$ Conductor $75712$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("u1")

sage: E.isogeny_class()

## Elliptic curves in class 75712.u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.u1 75712df1 $$[0, -1, 0, -178689, 30643105]$$ $$-226981/14$$ $$-38918682370899968$$ $$[]$$ $$599040$$ $$1.9378$$ $$\Gamma_0(N)$$-optimal
75712.u2 75712df2 $$[0, -1, 0, 524351, -1854769567]$$ $$5735339/537824$$ $$-1495100101960493170688$$ $$[]$$ $$2995200$$ $$2.7425$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 75712.2.a.u

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{5} + q^{7} - 2q^{9} - 5q^{11} + 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.