# Properties

 Label 12274.f Number of curves 4 Conductor 12274 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12274.f1")

sage: E.isogeny_class()

## Elliptic curves in class 12274.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12274.f1 12274d4 [1, 1, 0, -40800, 2175014]  82944
12274.f2 12274d3 [1, 1, 0, -37190, 2744672]  41472
12274.f3 12274d2 [1, 1, 0, -15530, -751252]  27648
12274.f4 12274d1 [1, 1, 0, -1090, -9036]  13824 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 12274.f have rank $$0$$.

## Modular form 12274.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 4q^{7} - q^{8} + q^{9} + 6q^{11} + 2q^{12} - 2q^{13} + 4q^{14} + q^{16} - q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 