# Properties

 Label 12274d Number of curves $4$ Conductor $12274$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 12274d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12274.f4 12274d1 $$[1, 1, 0, -1090, -9036]$$ $$3048625/1088$$ $$51185918528$$ $$$$ $$13824$$ $$0.75583$$ $$\Gamma_0(N)$$-optimal
12274.f3 12274d2 $$[1, 1, 0, -15530, -751252]$$ $$8805624625/2312$$ $$108770076872$$ $$$$ $$27648$$ $$1.1024$$
12274.f2 12274d3 $$[1, 1, 0, -37190, 2744672]$$ $$120920208625/19652$$ $$924545653412$$ $$$$ $$41472$$ $$1.3051$$
12274.f1 12274d4 $$[1, 1, 0, -40800, 2175014]$$ $$159661140625/48275138$$ $$2271146397606578$$ $$$$ $$82944$$ $$1.6517$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 12274d do not have complex multiplication.

## Modular form 12274.2.a.d

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 Cremona 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 Cremona labels. 