# Properties

 Label 16.0.308...281.2 Degree $16$ Signature $[0, 8]$ Discriminant $3.085\times 10^{19}$ Root discriminant $16.52$ Ramified primes $3, 7, 13$ Class number $1$ Class group trivial Galois group $\SL(2,3):C_2$ (as 16T60)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7)

gp: K = bnfinit(x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -7, -28, -9, 78, 112, -61, -183, 32, 183, 7, -101, -8, 31, 0, -5, 1]);

$$x^{16} - 5 x^{15} + 31 x^{13} - 8 x^{12} - 101 x^{11} + 7 x^{10} + 183 x^{9} + 32 x^{8} - 183 x^{7} - 61 x^{6} + 112 x^{5} + 78 x^{4} - 9 x^{3} - 28 x^{2} - 7 x + 7$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$30853268336830129281$$$$\medspace = 3^{8}\cdot 7^{8}\cdot 13^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $16.52$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $3, 7, 13$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $4$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{14} a^{14} + \frac{1}{14} a^{12} - \frac{3}{7} a^{11} + \frac{5}{14} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{14} a^{7} + \frac{3}{14} a^{6} + \frac{1}{14} a^{5} + \frac{3}{14} a^{4} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{201188162} a^{15} - \frac{2928463}{100594081} a^{14} + \frac{8434785}{201188162} a^{13} + \frac{28114571}{100594081} a^{12} - \frac{98515681}{201188162} a^{11} + \frac{25976444}{100594081} a^{10} - \frac{7398143}{100594081} a^{9} - \frac{59162769}{201188162} a^{8} - \frac{810261}{1991962} a^{7} - \frac{38192121}{201188162} a^{6} - \frac{20136635}{201188162} a^{5} - \frac{9431326}{100594081} a^{4} + \frac{87385559}{201188162} a^{3} + \frac{2754567}{14370583} a^{2} + \frac{9175725}{28741166} a + \frac{177689}{14370583}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$\frac{2412144}{14370583} a^{15} - \frac{14391461}{14370583} a^{14} + \frac{15876474}{14370583} a^{13} + \frac{49247605}{14370583} a^{12} - \frac{65201610}{14370583} a^{11} - \frac{117821061}{14370583} a^{10} + \frac{93665114}{14370583} a^{9} + \frac{169284546}{14370583} a^{8} - \frac{48574}{142283} a^{7} - \frac{146349640}{14370583} a^{6} - \frac{65202436}{14370583} a^{5} + \frac{89113643}{14370583} a^{4} + \frac{104995800}{14370583} a^{3} + \frac{10704428}{14370583} a^{2} + \frac{3108868}{14370583} a - \frac{5893873}{14370583}$$ (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$7145.924896370814$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 7145.924896370814 \cdot 1}{6\sqrt{30853268336830129281}}\approx 0.520829477092895$

## Galois group

$\SL(2,3):C_2$ (as 16T60):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 14 conjugacy class representatives for $\SL(2,3):C_2$ Character table for $\SL(2,3):C_2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 24 sibling: Deg 24

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ R ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 3.8.4.1x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] 7.3.2.3x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3} 7.3.2.3x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3} 1313.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 13.6.4.3x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
1.91.3t1.a.a$1$ $7 \cdot 13$ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.273.6t1.d.a$1$ $3 \cdot 7 \cdot 13$ 6.0.1851523947.1 $C_6$ (as 6T1) $0$ $-1$
1.91.3t1.a.b$1$ $7 \cdot 13$ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.273.6t1.d.b$1$ $3 \cdot 7 \cdot 13$ 6.0.1851523947.1 $C_6$ (as 6T1) $0$ $-1$
2.24843.24t21.a.a$2$ $3 \cdot 7^{2} \cdot 13^{2}$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
2.24843.24t21.a.b$2$ $3 \cdot 7^{2} \cdot 13^{2}$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.a$2$ $3 \cdot 7 \cdot 13$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.b$2$ $3 \cdot 7 \cdot 13$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.c$2$ $3 \cdot 7 \cdot 13$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.273.16t60.a.d$2$ $3 \cdot 7 \cdot 13$ 16.0.30853268336830129281.2 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 3.24843.6t6.a.a$3$ $3 \cdot 7^{2} \cdot 13^{2}$ 6.4.205724883.1 $A_4\times C_2$ (as 6T6) $1$ $1$
* 3.74529.4t4.a.a$3$ $3^{2} \cdot 7^{2} \cdot 13^{2}$ 4.0.74529.1 $A_4$ (as 4T4) $1$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.