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)

Related objects

Show commands: SageMath / Pari/GP / Magma

Normalizeddefining 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]);

$$x^{16} - 8x^{14} + 4x^{12} + 120x^{10} + 198x^{8} - 120x^{6} + 4x^{4} + 8x^{2} + 1$$

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: $$4668406261161555656704$$ 4668406261161555656704 $$\medspace = 2^{38}\cdot 19^{8}$$ 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: $$2$$, $$19$$ 2, 19 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 $$a$$)

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

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

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
$$2$$ Deg $16$$16$$1$$38 $$19$$ 19.4.0.1x^{4} - 2 x + 10$$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 *.