# Properties

 Label 25872cw Number of curves 4 Conductor 25872 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25872.ci1")

sage: E.isogeny_class()

## Elliptic curves in class 25872cw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25872.ci3 25872cw1 [0, 1, 0, -4328, 101940]  34560 $$\Gamma_0(N)$$-optimal
25872.ci4 25872cw2 [0, 1, 0, 3512, 437492]  69120
25872.ci1 25872cw3 [0, 1, 0, -63128, -6102636]  103680
25872.ci2 25872cw4 [0, 1, 0, -31768, -12136300]  207360

## Rank

sage: E.rank()

The elliptic curves in class 25872cw have rank $$1$$.

## Modular form 25872.2.a.ci

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} + q^{11} + 4q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 