# Properties

 Label 257754.bb Number of curves $6$ Conductor $257754$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("257754.bb1")

sage: E.isogeny_class()

## Elliptic curves in class 257754.bb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
257754.bb1 257754bb5 [1, 0, 1, -4952416052, 134144366143754] [2] 212336640
257754.bb2 257754bb3 [1, 0, 1, -310114892, 2087610425930] [2, 2] 106168320
257754.bb3 257754bb6 [1, 0, 1, -105630052, 4799570167946] [2] 212336640
257754.bb4 257754bb2 [1, 0, 1, -32751372, -18133417910] [2, 2] 53084160
257754.bb5 257754bb1 [1, 0, 1, -25358092, -49090559926] [2] 26542080 $$\Gamma_0(N)$$-optimal
257754.bb6 257754bb4 [1, 0, 1, 126319668, -142590599606] [2] 106168320

## Rank

sage: E.rank()

The elliptic curves in class 257754.bb have rank $$1$$.

## Modular form 257754.2.a.bb

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{14} - 2q^{15} + q^{16} + q^{17} - q^{18} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.