# Properties

 Label 2475.i Number of curves $4$ Conductor $2475$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("i1")

E.isogeny_class()

## Elliptic curves in class 2475.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.i1 2475j3 $$[1, -1, 0, -13317, -588034]$$ $$22930509321/6875$$ $$78310546875$$ $$[2]$$ $$3072$$ $$1.0684$$
2475.i2 2475j4 $$[1, -1, 0, -6567, 201716]$$ $$2749884201/73205$$ $$833850703125$$ $$[2]$$ $$3072$$ $$1.0684$$
2475.i3 2475j2 $$[1, -1, 0, -942, -6409]$$ $$8120601/3025$$ $$34456640625$$ $$[2, 2]$$ $$1536$$ $$0.72179$$
2475.i4 2475j1 $$[1, -1, 0, 183, -784]$$ $$59319/55$$ $$-626484375$$ $$[2]$$ $$768$$ $$0.37521$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2475.2.a.i

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3 q^{8} + q^{11} - 2 q^{13} - q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.