Properties

Label 16.0.466...704.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.668\times 10^{21}$
Root discriminant \(22.61\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $\SL(2,3):C_2$ (as 16T60)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 1)
 
gp: K = bnfinit(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 8, 0, 4, 0, -120, 0, 198, 0, 120, 0, 4, 0, -8, 0, 1]);
 

\( x^{16} - 8x^{14} + 4x^{12} + 120x^{10} + 198x^{8} - 120x^{6} + 4x^{4} + 8x^{2} + 1 \) Copy content Toggle raw display

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:   \(4668406261161555656704\) \(\medspace = 2^{38}\cdot 19^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(22.61\)
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:   \(2\), \(19\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{16}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}+\frac{1}{16}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{5}{16}a+\frac{1}{4}$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{3}{16}a^{2}+\frac{1}{4}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}+\frac{9}{32}a^{3}+\frac{7}{32}a^{2}+\frac{7}{32}a-\frac{9}{32}$, $\frac{1}{64}a^{12}-\frac{1}{32}a^{10}+\frac{1}{64}a^{8}+\frac{1}{16}a^{6}+\frac{15}{64}a^{4}-\frac{5}{32}a^{2}+\frac{31}{64}$, $\frac{1}{64}a^{13}-\frac{1}{32}a^{10}-\frac{1}{64}a^{9}-\frac{1}{32}a^{8}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{5}{64}a^{5}-\frac{1}{16}a^{4}-\frac{1}{8}a^{3}+\frac{7}{32}a^{2}+\frac{29}{64}a-\frac{9}{32}$, $\frac{1}{320}a^{14}+\frac{9}{320}a^{10}+\frac{1}{160}a^{8}-\frac{1}{320}a^{6}-\frac{17}{80}a^{4}+\frac{27}{64}a^{2}+\frac{69}{160}$, $\frac{1}{640}a^{15}-\frac{1}{640}a^{14}-\frac{1}{128}a^{13}-\frac{1}{128}a^{12}-\frac{1}{640}a^{11}+\frac{1}{640}a^{10}+\frac{17}{640}a^{9}-\frac{7}{640}a^{8}+\frac{19}{640}a^{7}+\frac{61}{640}a^{6}-\frac{23}{640}a^{5}-\frac{87}{640}a^{4}-\frac{47}{128}a^{3}-\frac{1}{128}a^{2}-\frac{77}{640}a-\frac{53}{640}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:   \( \frac{19}{80} a^{14} - \frac{61}{32} a^{12} + \frac{81}{80} a^{10} + \frac{4541}{160} a^{8} + \frac{3701}{80} a^{6} - \frac{4499}{160} a^{4} + \frac{75}{16} a^{2} + \frac{159}{160} \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $a$, $\frac{1}{128}a^{15}+\frac{33}{640}a^{14}-\frac{1}{128}a^{13}-\frac{49}{128}a^{12}-\frac{49}{128}a^{11}-\frac{13}{640}a^{10}+\frac{125}{128}a^{9}+\frac{3981}{640}a^{8}+\frac{1043}{128}a^{7}+\frac{8827}{640}a^{6}+\frac{1621}{128}a^{5}+\frac{721}{640}a^{4}-\frac{123}{128}a^{3}-\frac{139}{128}a^{2}-\frac{9}{128}a-\frac{561}{640}$, $\frac{117}{640}a^{15}+\frac{89}{640}a^{14}-\frac{183}{128}a^{13}-\frac{143}{128}a^{12}+\frac{303}{640}a^{11}+\frac{391}{640}a^{10}+\frac{14099}{640}a^{9}+\frac{10583}{640}a^{8}+\frac{25703}{640}a^{7}+\frac{17291}{640}a^{6}-\frac{9481}{640}a^{5}-\frac{9977}{640}a^{4}-\frac{343}{128}a^{3}+\frac{633}{128}a^{2}+\frac{81}{640}a+\frac{517}{640}$, $\frac{239}{640}a^{15}+\frac{127}{640}a^{14}-\frac{391}{128}a^{13}-\frac{207}{128}a^{12}+\frac{1321}{640}a^{11}+\frac{653}{640}a^{10}+\frac{28343}{640}a^{9}+\frac{15219}{640}a^{8}+\frac{42221}{640}a^{7}+\frac{22853}{640}a^{6}-\frac{34657}{640}a^{5}-\frac{19921}{640}a^{4}+\frac{2135}{128}a^{3}+\frac{299}{128}a^{2}-\frac{603}{640}a+\frac{1521}{640}$, $\frac{139}{320}a^{15}+\frac{3}{4}a^{14}-\frac{29}{8}a^{13}-\frac{97}{16}a^{12}+\frac{951}{320}a^{11}+\frac{7}{2}a^{10}+\frac{8199}{160}a^{9}+\frac{359}{4}a^{8}+\frac{21821}{320}a^{7}+141a^{6}-\frac{6213}{80}a^{5}-\frac{1637}{16}a^{4}+\frac{1645}{64}a^{3}+\frac{41}{4}a^{2}-\frac{509}{160}a+\frac{43}{8}$, $\frac{3}{40}a^{15}-\frac{7}{64}a^{14}-\frac{37}{64}a^{13}+\frac{53}{64}a^{12}+\frac{23}{160}a^{11}-\frac{5}{64}a^{10}+\frac{2863}{320}a^{9}-\frac{845}{64}a^{8}+\frac{1399}{80}a^{7}-\frac{1741}{64}a^{6}-\frac{807}{320}a^{5}+\frac{111}{64}a^{4}+\frac{95}{32}a^{3}+\frac{49}{64}a^{2}+\frac{417}{320}a+\frac{17}{64}$, $\frac{51}{160}a^{15}+\frac{1}{80}a^{14}-\frac{161}{64}a^{13}-\frac{3}{32}a^{12}+\frac{41}{40}a^{11}-\frac{1}{80}a^{10}+\frac{12219}{320}a^{9}+\frac{259}{160}a^{8}+\frac{10779}{160}a^{7}+\frac{259}{80}a^{6}-\frac{9131}{320}a^{5}-\frac{301}{160}a^{4}+\frac{29}{16}a^{3}-\frac{55}{16}a^{2}+\frac{101}{320}a-\frac{79}{160}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 60925.785768593894 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 60925.785768593894 \cdot 1}{4\sqrt{4668406261161555656704}}\approx 0.541496648727078$

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

\(\Q(\sqrt{-1}) \), 4.0.23104.1, 8.0.2135179264.1

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 R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ R ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(2\) Copy content Toggle raw display Deg $16$$16$$1$$38$
\(19\) Copy content Toggle raw display 19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.12.8.1$x^{12} - 114 x^{9} + 4332 x^{6} - 54872 x^{3} + 130321000$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.76.6t1.a.a$1$ $ 2^{2} \cdot 19 $ 6.0.8340544.1 $C_6$ (as 6T1) $0$ $-1$
1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.76.6t1.a.b$1$ $ 2^{2} \cdot 19 $ 6.0.8340544.1 $C_6$ (as 6T1) $0$ $-1$
2.23104.24t21.a.a$2$ $ 2^{6} \cdot 19^{2}$ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
2.23104.24t21.a.b$2$ $ 2^{6} \cdot 19^{2}$ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.1216.16t60.a.a$2$ $ 2^{6} \cdot 19 $ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.1216.16t60.a.b$2$ $ 2^{6} \cdot 19 $ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.1216.16t60.a.c$2$ $ 2^{6} \cdot 19 $ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 2.1216.16t60.a.d$2$ $ 2^{6} \cdot 19 $ 16.0.4668406261161555656704.1 $\SL(2,3):C_2$ (as 16T60) $0$ $0$
* 3.23104.6t6.a.a$3$ $ 2^{6} \cdot 19^{2}$ 6.4.8340544.1 $A_4\times C_2$ (as 6T6) $1$ $1$
* 3.23104.4t4.b.a$3$ $ 2^{6} \cdot 19^{2}$ 4.0.23104.1 $A_4$ (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 *.