# Properties

 Label 18496.i Number of curves $4$ Conductor $18496$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("18496.i1")

sage: E.isogeny_class()

## Elliptic curves in class 18496.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
18496.i1 18496c3 [0, 0, 0, -1677356, -836153296]  147456
18496.i2 18496c2 [0, 0, 0, -105196, -12970320] [2, 2] 73728
18496.i3 18496c4 [0, 0, 0, -12716, -34980560]  147456
18496.i4 18496c1 [0, 0, 0, -12716, 235824]  36864 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 18496.i have rank $$1$$.

## Modular form 18496.2.a.i

sage: E.q_eigenform(10)

$$q - 2q^{5} - 4q^{7} - 3q^{9} + 2q^{13} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 