# Properties

 Label 475a Number of curves $3$ Conductor $475$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 475a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
475.b3 475a1 $$[0, -1, 1, 17, -7]$$ $$32768/19$$ $$-296875$$ $$[]$$ $$36$$ $$-0.26045$$ $$\Gamma_0(N)$$-optimal
475.b2 475a2 $$[0, -1, 1, -233, -1382]$$ $$-89915392/6859$$ $$-107171875$$ $$[]$$ $$108$$ $$0.28885$$
475.b1 475a3 $$[0, -1, 1, -19233, -1020257]$$ $$-50357871050752/19$$ $$-296875$$ $$[]$$ $$324$$ $$0.83816$$

## Rank

sage: E.rank()

The elliptic curves in class 475a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 475a do not have complex multiplication.

## Modular form475.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} + q^{7} + q^{9} + 3q^{11} - 4q^{12} + 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 