# Properties

 Label 405720k Number of curves $2$ Conductor $405720$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 405720k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405720.k1 405720k1 $$[0, 0, 0, -4263, 29498]$$ $$10536048/5635$$ $$4582325018880$$ $$$$ $$737280$$ $$1.1207$$ $$\Gamma_0(N)$$-optimal
405720.k2 405720k2 $$[0, 0, 0, 16317, 231182]$$ $$147704148/92575$$ $$-301124215526400$$ $$$$ $$1474560$$ $$1.4673$$

## Rank

sage: E.rank()

The elliptic curves in class 405720k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405720k do not have complex multiplication.

## Modular form 405720.2.a.k

sage: E.q_eigenform(10)

$$q - q^{5} - 4q^{11} - 4q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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 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. 