# Properties

 Label 2475g Number of curves $4$ Conductor $2475$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 2475g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.g3 2475g1 $$[1, -1, 0, -1467, 21816]$$ $$30664297/297$$ $$3383015625$$ $$$$ $$1536$$ $$0.64825$$ $$\Gamma_0(N)$$-optimal
2475.g2 2475g2 $$[1, -1, 0, -2592, -15309]$$ $$169112377/88209$$ $$1004755640625$$ $$[2, 2]$$ $$3072$$ $$0.99482$$
2475.g1 2475g3 $$[1, -1, 0, -32967, -2293434]$$ $$347873904937/395307$$ $$4502793796875$$ $$$$ $$6144$$ $$1.3414$$
2475.g4 2475g4 $$[1, -1, 0, 9783, -126684]$$ $$9090072503/5845851$$ $$-66587896546875$$ $$$$ $$6144$$ $$1.3414$$

## Rank

sage: E.rank()

The elliptic curves in class 2475g have rank $$0$$.

## Complex multiplication

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

## Modular form2475.2.a.g

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 