# Properties

 Label 6253.b Number of curves 3 Conductor 6253 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6253.b1")

sage: E.isogeny_class()

## Elliptic curves in class 6253.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6253.b1 6253a3 [0, 1, 1, -316593, -68670260] [] 12960
6253.b2 6253a2 [0, 1, 1, -3943, -93609] [] 4320
6253.b3 6253a1 [0, 1, 1, -563, 4918] [] 1440 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6253.b have rank $$1$$.

## Modular form6253.2.a.b

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} + q^{7} - 2q^{9} - 3q^{11} - 2q^{12} + 4q^{16} + 6q^{17} - 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 