# Properties

 Label 475c Number of curves $2$ Conductor $475$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("c1")

E.isogeny_class()

## Elliptic curves in class 475c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
475.a2 475c1 $$[1, -1, 1, 0, 2]$$ $$27/19$$ $$-2375$$ $$$$ $$16$$ $$-0.67323$$ $$\Gamma_0(N)$$-optimal
475.a1 475c2 $$[1, -1, 1, -25, 52]$$ $$13312053/361$$ $$45125$$ $$$$ $$32$$ $$-0.32666$$

## Rank

sage: E.rank()

The elliptic curves in class 475c have rank $$1$$.

## Complex multiplication

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

## Modular form475.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} + 4 q^{17} + 3 q^{18} + q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 