# Properties

 Label 309442.a Number of curves $2$ Conductor $309442$ CM no Rank $2$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 309442.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
309442.a1 309442a2 $$[1, 0, 1, -581906, -160003164]$$ $$24553362849625/1755162752$$ $$1557713403154090112$$ $$$$ $$6773760$$ $$2.2377$$
309442.a2 309442a1 $$[1, 0, 1, 33134, -10917468]$$ $$4533086375/60669952$$ $$-53844805726093312$$ $$$$ $$3386880$$ $$1.8912$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 309442.a have rank $$2$$.

## Complex multiplication

The elliptic curves in class 309442.a do not have complex multiplication.

## Modular form 309442.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + q^{7} - q^{8} + q^{9} - 4q^{11} - 2q^{12} - q^{14} + q^{16} - 6q^{17} - q^{18} - 6q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 