# Properties

 Label 57475.g Number of curves $3$ Conductor $57475$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 57475.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57475.g1 57475h3 $$[0, -1, 1, -2327233, 1367270618]$$ $$-50357871050752/19$$ $$-525932171875$$ $$[]$$ $$437400$$ $$2.0371$$
57475.g2 57475h2 $$[0, -1, 1, -28233, 1951993]$$ $$-89915392/6859$$ $$-189861514046875$$ $$[]$$ $$145800$$ $$1.4878$$
57475.g3 57475h1 $$[0, -1, 1, 2017, 868]$$ $$32768/19$$ $$-525932171875$$ $$[]$$ $$48600$$ $$0.93849$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 57475.g have rank $$0$$.

## Complex multiplication

The elliptic curves in class 57475.g do not have complex multiplication.

## Modular form 57475.2.a.g

sage: E.q_eigenform(10)

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