# Properties

 Label 4025.e Number of curves $4$ Conductor $4025$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 4025.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4025.e1 4025a3 $$[1, -1, 0, -3092, 66941]$$ $$209267191953/55223$$ $$862859375$$ $$$$ $$2560$$ $$0.69909$$
4025.e2 4025a2 $$[1, -1, 0, -217, 816]$$ $$72511713/25921$$ $$405015625$$ $$[2, 2]$$ $$1280$$ $$0.35251$$
4025.e3 4025a1 $$[1, -1, 0, -92, -309]$$ $$5545233/161$$ $$2515625$$ $$$$ $$640$$ $$0.0059379$$ $$\Gamma_0(N)$$-optimal
4025.e4 4025a4 $$[1, -1, 0, 658, 5191]$$ $$2014698447/1958887$$ $$-30607609375$$ $$$$ $$2560$$ $$0.69909$$

## Rank

sage: E.rank()

The elliptic curves in class 4025.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 4025.e do not have complex multiplication.

## Modular form4025.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - q^{7} - 3 q^{8} - 3 q^{9} + 4 q^{11} - 6 q^{13} - q^{14} - q^{16} + 2 q^{17} - 3 q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 