# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("4025.f1")

sage: E.isogeny_class()

## Elliptic curves in class 4025.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4025.f1 4025c4 [1, -1, 0, -1052192, 415686091]  34560
4025.f2 4025c3 [1, -1, 0, -137942, -10015159]  34560
4025.f3 4025c2 [1, -1, 0, -66067, 6444216] [2, 2] 17280
4025.f4 4025c1 [1, -1, 0, 58, 294591]  8640 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form4025.2.a.f

sage: E.q_eigenform(10)

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

## 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. 