LMFDB - Number field 12.6.122023936000000.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1)

gp:K = bnfinit(y^12 - 2*y^10 - 4*y^9 - 9*y^8 + 2*y^7 + 20*y^6 + 22*y^5 + 13*y^4 - 2*y^3 - 9*y^2 - 6*y - 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1)

$x^{12} - 2x^{10} - 4x^{9} - 9x^{8} + 2x^{7} + 20x^{6} + 22x^{5} + 13x^{4} - 2x^{3} - 9x^{2} - 6x - 1$ x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$12$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[6, 3]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-122023936000000$ -122023936000000 $\medspace = -\,2^{18}\cdot 5^{6}\cdot 31^{3}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$14.92$	sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:(1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{3/2}5^{1/2}31^{5/6}\approx 110.61942767882832$
Ramified primes:	$2$ , $5$ , $31$ 2, 5, 31	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-31})$
$\Aut(K/\Q)$ :	$C_1$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$ .
This is not a CM field.
This field has no CM subfields.

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

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ , $\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$ , $\frac{1}{325}a^{11}+\frac{21}{325}a^{10}-\frac{146}{325}a^{9}-\frac{3}{65}a^{8}+\frac{66}{325}a^{7}+\frac{153}{325}a^{6}+\frac{113}{325}a^{5}+\frac{24}{65}a^{4}+\frac{128}{325}a^{3}-\frac{44}{325}a^{2}+\frac{107}{325}a+\frac{96}{325}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/5*a^10 + 2/5*a^9 + 1/5*a^8 + 1/5*a^7 + 2/5*a^6 - 2/5*a^4 - 2/5*a^3 + 1/5*a^2 + 2/5*a - 1/5, 1/325*a^11 + 21/325*a^10 - 146/325*a^9 - 3/65*a^8 + 66/325*a^7 + 153/325*a^6 + 113/325*a^5 + 24/65*a^4 + 128/325*a^3 - 44/325*a^2 + 107/325*a + 96/325

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

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

Class group and class number

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$8$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$-1$ -1 (order $2$ )	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$\frac{22}{65}a^{11}-\frac{19}{65}a^{10}-\frac{14}{65}a^{9}-\frac{96}{65}a^{8}-\frac{134}{65}a^{7}+\frac{129}{65}a^{6}+\frac{276}{65}a^{5}+\frac{417}{65}a^{4}+\frac{138}{65}a^{3}-\frac{149}{65}a^{2}-\frac{168}{65}a-\frac{72}{65}$ , $a$ , $\frac{226}{325}a^{11}-\frac{129}{325}a^{10}-\frac{496}{325}a^{9}-\frac{93}{65}a^{8}-\frac{1659}{325}a^{7}+\frac{1753}{325}a^{6}+\frac{4088}{325}a^{5}+\frac{289}{65}a^{4}+\frac{653}{325}a^{3}-\frac{1819}{325}a^{2}-\frac{1168}{325}a-\frac{79}{325}$ , $\frac{236}{325}a^{11}-\frac{179}{325}a^{10}-\frac{201}{325}a^{9}-\frac{35}{13}a^{8}-\frac{1584}{325}a^{7}+\frac{1138}{325}a^{6}+\frac{2943}{325}a^{5}+\frac{698}{65}a^{4}+\frac{2128}{325}a^{3}+\frac{81}{325}a^{2}-\frac{1268}{325}a-\frac{484}{325}$ , $\frac{212}{325}a^{11}-\frac{293}{325}a^{10}-\frac{142}{325}a^{9}-\frac{18}{13}a^{8}-\frac{1153}{325}a^{7}+\frac{2146}{325}a^{6}+\frac{1856}{325}a^{5}+\frac{96}{65}a^{4}+\frac{876}{325}a^{3}-\frac{1073}{325}a^{2}-\frac{456}{325}a+\frac{72}{325}$ , $\frac{141}{325}a^{11}-\frac{29}{325}a^{10}-\frac{241}{325}a^{9}-\frac{111}{65}a^{8}-\frac{1159}{325}a^{7}+\frac{318}{325}a^{6}+\frac{2608}{325}a^{5}+\frac{110}{13}a^{4}+\frac{1603}{325}a^{3}+\frac{231}{325}a^{2}-\frac{968}{325}a-\frac{374}{325}$ , $\frac{357}{325}a^{11}-\frac{498}{325}a^{10}-\frac{187}{325}a^{9}-\frac{40}{13}a^{8}-\frac{1658}{325}a^{7}+\frac{3531}{325}a^{6}+\frac{3291}{325}a^{5}+\frac{391}{65}a^{4}-\frac{64}{325}a^{3}-\frac{2253}{325}a^{2}-\frac{1191}{325}a+\frac{17}{325}$ , $\frac{9}{25}a^{11}-\frac{11}{25}a^{10}-\frac{14}{25}a^{9}-\frac{2}{5}a^{8}-\frac{56}{25}a^{7}+\frac{102}{25}a^{6}+\frac{117}{25}a^{5}-\frac{4}{5}a^{4}+\frac{27}{25}a^{3}-\frac{71}{25}a^{2}-\frac{37}{25}a+\frac{14}{25}$ 22/65a^11 - 19/65a^10 - 14/65a^9 - 96/65a^8 - 134/65a^7 + 129/65a^6 + 276/65a^5 + 417/65a^4 + 138/65a^3 - 149/65a^2 - 168/65a - 72/65, a, 226/325a^11 - 129/325a^10 - 496/325a^9 - 93/65a^8 - 1659/325a^7 + 1753/325a^6 + 4088/325a^5 + 289/65a^4 + 653/325a^3 - 1819/325a^2 - 1168/325a - 79/325, 236/325a^11 - 179/325a^10 - 201/325a^9 - 35/13a^8 - 1584/325a^7 + 1138/325a^6 + 2943/325a^5 + 698/65a^4 + 2128/325a^3 + 81/325a^2 - 1268/325a - 484/325, 212/325a^11 - 293/325a^10 - 142/325a^9 - 18/13a^8 - 1153/325a^7 + 2146/325a^6 + 1856/325a^5 + 96/65a^4 + 876/325a^3 - 1073/325a^2 - 456/325a + 72/325, 141/325a^11 - 29/325a^10 - 241/325a^9 - 111/65a^8 - 1159/325a^7 + 318/325a^6 + 2608/325a^5 + 110/13a^4 + 1603/325a^3 + 231/325a^2 - 968/325a - 374/325, 357/325a^11 - 498/325a^10 - 187/325a^9 - 40/13a^8 - 1658/325a^7 + 3531/325a^6 + 3291/325a^5 + 391/65a^4 - 64/325a^3 - 2253/325a^2 - 1191/325a + 17/325, 9/25a^11 - 11/25a^10 - 14/25a^9 - 2/5a^8 - 56/25a^7 + 102/25a^6 + 117/25a^5 - 4/5a^4 + 27/25a^3 - 71/25a^2 - 37/25*a + 14/25	comment:Fundamental units sage:UK.fundamental_units() gp:K.fu magma:[K\|fUK(g): g in Generators(UK)]; oscar:[K(fUK(a)) for a in gens(UK)]
Regulator:	$280.158137734$	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 280.158137734 \cdot 1}{2\cdot\sqrt{122023936000000}}\cr\approx \mathstrut & 0.201312299938 \end{aligned}$

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$S_3\wr C_2^2$ (as 12T261):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 5184

The 45 conjugacy class representatives for

S_3\wr C_2^2

Character table for

S_3\wr C_2^2

Intermediate fields

\Q(\sqrt{5})

\Q(\sqrt{2})

\Q(\sqrt{10})

\Q(\sqrt{2}, \sqrt{5})

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

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(b)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 12 siblings:	data not computed
Degree 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$	R	${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$	${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$	${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$	R	${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$	${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.6.2.18a1.17	$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$	$2$	$6$	$18$	$C_6\times C_2$	$[3]^{6}$
$5$ 5	5.2.2.2a1.2	$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$5$ 5	5.4.2.4a1.2	$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$31$ 31	$\Q_{31}$	$x + 28$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{31}$	$x + 28$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{31}$	$x + 28$	$1$	$1$	$0$	Trivial	$[\ ]$
	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.1.3.2a1.3	$x^{3} + 279$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Normalized defining polynomial

Invariants

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

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Sibling fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	12.6.122023936000000.1

Degree	$12$
Signature	$[6, 3]$
Discriminant	$-1.220\times 10^{14}$
Root discriminant	$14.92$
Ramified primes	$2,5,31$
Class number	$1$
Class group	trivial
Galois group	$S_3\wr C_2^2$ (as 12T261)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Sibling fields

Local algebras for ramified primes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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