# Properties

 Label 13860.k Number of curves $4$ Conductor $13860$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 13860.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.k1 13860c3 $$[0, 0, 0, -9288, 27837]$$ $$281370820608/161767375$$ $$50945075874000$$ $$[2]$$ $$31104$$ $$1.3198$$
13860.k2 13860c1 $$[0, 0, 0, -6648, 208633]$$ $$75216478666752/326095$$ $$140873040$$ $$[6]$$ $$10368$$ $$0.77046$$ $$\Gamma_0(N)$$-optimal
13860.k3 13860c2 $$[0, 0, 0, -6543, 215542]$$ $$-4481782160112/310023175$$ $$-2142880185600$$ $$[6]$$ $$20736$$ $$1.1170$$
13860.k4 13860c4 $$[0, 0, 0, 37017, 222318]$$ $$1113258734352/648484375$$ $$-3267614196000000$$ $$[2]$$ $$62208$$ $$1.6663$$

## Rank

sage: E.rank()

The elliptic curves in class 13860.k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 13860.k do not have complex multiplication.

## Modular form 13860.2.a.k

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} + q^{11} + 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.