# Properties

 Label 123981h Number of curves $6$ Conductor $123981$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("123981.e1")

sage: E.isogeny_class()

## Elliptic curves in class 123981h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123981.e5 123981h1 [1, 1, 1, -6942, 316458] [2] 294912 $$\Gamma_0(N)$$-optimal
123981.e4 123981h2 [1, 1, 1, -123987, 16749576] [2, 2] 589824
123981.e3 123981h3 [1, 1, 1, -136992, 13004136] [2, 2] 1179648
123981.e1 123981h4 [1, 1, 1, -1983702, 1074555468] [2] 1179648
123981.e6 123981h5 [1, 1, 1, 387543, 89166618] [2] 2359296
123981.e2 123981h6 [1, 1, 1, -869607, -302606406] [2] 2359296

## Rank

sage: E.rank()

The elliptic curves in class 123981h have rank $$1$$.

## Modular form 123981.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + 3q^{8} + q^{9} - 2q^{10} + q^{11} + q^{12} + q^{13} - 2q^{15} - q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.