# Properties

 Label 2475.e Number of curves $2$ Conductor $2475$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 2475.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.e1 2475k1 $$[0, 0, 1, -5250, -153594]$$ $$-56197120/3267$$ $$-930329296875$$ $$[]$$ $$2880$$ $$1.0528$$ $$\Gamma_0(N)$$-optimal
2475.e2 2475k2 $$[0, 0, 1, 28500, -271719]$$ $$8990228480/5314683$$ $$-1513439026171875$$ $$$$ $$8640$$ $$1.6021$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2475.2.a.e

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. 