# Properties

 Label 214774a Number of curves $2$ Conductor $214774$ CM no Rank $2$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 214774a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
214774.k2 214774a1 $$[1, 0, 0, 48657, 97321]$$ $$86058173375/49827568$$ $$-7376268325587952$$ $$$$ $$1419264$$ $$1.7342$$ $$\Gamma_0(N)$$-optimal
214774.k1 214774a2 $$[1, 0, 0, -194683, 730005]$$ $$5512402554625/3188422748$$ $$472000996008002972$$ $$$$ $$2838528$$ $$2.0808$$

## Rank

sage: E.rank()

The elliptic curves in class 214774a have rank $$2$$.

## Complex multiplication

The elliptic curves in class 214774a do not have complex multiplication.

## Modular form 214774.2.a.a

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 