# Properties

 Label 25410bt Number of curves $4$ Conductor $25410$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bt1")

sage: E.isogeny_class()

## Elliptic curves in class 25410bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25410.bu3 25410bt1 [1, 1, 1, -426, 1959] [2] 20480 $$\Gamma_0(N)$$-optimal
25410.bu2 25410bt2 [1, 1, 1, -2846, -58057] [2, 2] 40960
25410.bu4 25410bt3 [1, 1, 1, 784, -191641] [2] 81920
25410.bu1 25410bt4 [1, 1, 1, -45196, -3717097] [2] 81920

## Rank

sage: E.rank()

The elliptic curves in class 25410bt have rank $$0$$.

## Complex multiplication

The elliptic curves in class 25410bt do not have complex multiplication.

## Modular form 25410.2.a.bt

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} + q^{14} + q^{15} + q^{16} + 6q^{17} + q^{18} + O(q^{20})$$

## 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.