# Properties

 Label 303450.k Number of curves $4$ Conductor $303450$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 303450.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303450.k1 303450k3 $$[1, 1, 0, -36695925, 85545487125]$$ $$14489843500598257/6246072$$ $$2355703029358875000$$ $$$$ $$28311552$$ $$2.8679$$
303450.k2 303450k4 $$[1, 1, 0, -4905925, -2212830875]$$ $$34623662831857/14438442312$$ $$5445451524350305125000$$ $$$$ $$28311552$$ $$2.8679$$
303450.k3 303450k2 $$[1, 1, 0, -2304925, 1321928125]$$ $$3590714269297/73410624$$ $$27686781283329000000$$ $$[2, 2]$$ $$14155776$$ $$2.5213$$
303450.k4 303450k1 $$[1, 1, 0, 7075, 61888125]$$ $$103823/4386816$$ $$-1654485529536000000$$ $$$$ $$7077888$$ $$2.1747$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 303450.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 303450.k do not have complex multiplication.

## Modular form 303450.2.a.k

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 