# Properties

 Label 2475.f Number of curves $2$ Conductor $2475$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 2475.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.f1 2475i1 $$[0, 0, 1, -210, -1229]$$ $$-56197120/3267$$ $$-59541075$$ $$[]$$ $$576$$ $$0.24810$$ $$\Gamma_0(N)$$-optimal
2475.f2 2475i2 $$[0, 0, 1, 1140, -2174]$$ $$8990228480/5314683$$ $$-96860097675$$ $$[]$$ $$1728$$ $$0.79740$$

## Rank

sage: E.rank()

The elliptic curves in class 2475.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2475.f do not have complex multiplication.

## Modular form2475.2.a.f

sage: E.q_eigenform(10)

$$q - 2 q^{4} + q^{7} + q^{11} + q^{13} + 4 q^{16} - 6 q^{17} - 7 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 