# Properties

 Label 18491.a Number of curves 3 Conductor 18491 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("18491.a1")

sage: E.isogeny_class()

## Elliptic curves in class 18491.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
18491.a1 18491a3 [0, 1, 1, -13145980, -18350221122] [] 340000
18491.a2 18491a2 [0, 1, 1, -17370, -1601822] [] 68000
18491.a3 18491a1 [0, 1, 1, -560, 11938] [] 13600 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 18491.a have rank $$1$$.

## Modular form 18491.2.a.a

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. 