# Properties

 Label 9251.d Number of curves 3 Conductor 9251 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("9251.d1")

sage: E.isogeny_class()

## Elliptic curves in class 9251.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
9251.d1 9251a3 [0, 1, 1, -6576900, -6494214623] [] 126000
9251.d2 9251a2 [0, 1, 1, -8690, -567683] [] 25200
9251.d3 9251a1 [0, 1, 1, -280, 4197] [] 5040 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 9251.d have rank $$1$$.

## Modular form9251.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 