# Properties

 Label 12.0.33171021564453125.1 Degree $12$ Signature $[0, 6]$ Discriminant $3.317\times 10^{16}$ Root discriminant $$23.81$$ Ramified primes $5,19$ Class number $13$ Class group [13] Galois group $C_{12}$ (as 12T1)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)

gp: K = bnfinit(y^12 - y^11 + 7*y^10 - 6*y^9 + 41*y^8 - 62*y^7 + 266*y^6 - 351*y^5 + 1513*y^4 - 1757*y^3 + 2107*y^2 - 2058*y + 2401, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)

$$x^{12} - x^{11} + 7 x^{10} - 6 x^{9} + 41 x^{8} - 62 x^{7} + 266 x^{6} - 351 x^{5} + 1513 x^{4} + \cdots + 2401$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

## Invariants

 Degree: $12$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K);  oscar: degree(K) Signature: $[0, 6]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$33171021564453125$$ 33171021564453125 $$\medspace = 5^{9}\cdot 19^{8}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$23.81$$ 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: $5^{3/4}19^{2/3}\approx 23.8083831956814$ Ramified primes: $$5$$, $$19$$ 5, 19 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{5})$$ $\card{ \Gal(K/\Q) }$: $12$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is Galois and abelian over $\Q$. Conductor: $$95=5\cdot 19$$ Dirichlet character group: $\lbrace$$\chi_{95}(64,·), \chi_{95}(1,·), \chi_{95}(68,·), \chi_{95}(39,·), \chi_{95}(11,·), \chi_{95}(77,·), \chi_{95}(49,·), \chi_{95}(83,·), \chi_{95}(87,·), \chi_{95}(7,·), \chi_{95}(26,·), \chi_{95}(58,·)$$\rbrace$ This is a CM field. Reflex fields: $$\Q(\zeta_{5})$$$^{2}$, 12.0.33171021564453125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{360437}a^{9}+\frac{60066}{360437}a^{8}+\frac{211}{51491}a^{7}-\frac{1511}{360437}a^{6}+\frac{70398}{360437}a^{5}-\frac{60325}{360437}a^{4}-\frac{8540}{51491}a^{3}-\frac{20595}{360437}a^{2}-\frac{39486}{360437}a-\frac{20301}{51491}$, $\frac{1}{2523059}a^{10}-\frac{1}{2523059}a^{9}-\frac{52716}{360437}a^{8}+\frac{307469}{2523059}a^{7}+\frac{1511}{2523059}a^{6}-\frac{10107}{2523059}a^{5}-\frac{50229}{360437}a^{4}+\frac{111271}{2523059}a^{3}+\frac{1101906}{2523059}a^{2}-\frac{53206}{360437}a+\frac{8828}{51491}$, $\frac{1}{17661413}a^{11}-\frac{1}{17661413}a^{10}+\frac{1}{2523059}a^{9}-\frac{6839769}{17661413}a^{8}+\frac{2584889}{17661413}a^{7}-\frac{1777}{17661413}a^{6}+\frac{422222}{2523059}a^{5}-\frac{2533406}{17661413}a^{4}+\frac{2773982}{17661413}a^{3}+\frac{236860}{2523059}a^{2}-\frac{51204}{360437}a-\frac{10052}{51491}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: No Index: Not computed Inessential primes: $11$

## Class group and class number

$C_{13}$, which has order $13$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

## Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

 Rank: $5$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$-\frac{41}{360437} a^{11} - \frac{33}{32767} a^{6} + \frac{294302}{360437} a$$ -(41)/(360437)*a^(11) - (33)/(32767)*a^(6) + (294302)/(360437)*a  (order $10$) 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: $\frac{83}{2523059}a^{11}+\frac{204}{360437}a^{10}-\frac{1224}{2523059}a^{9}+\frac{8364}{2523059}a^{8}-\frac{809}{360437}a^{7}+\frac{8115}{360437}a^{6}-\frac{71604}{2523059}a^{5}+\frac{308652}{2523059}a^{4}-\frac{51204}{360437}a^{3}+\frac{2140312}{2523059}a^{2}-\frac{354278}{360437}a+\frac{9996}{51491}$, $\frac{82}{360437}a^{11}+\frac{66}{32767}a^{6}-\frac{228167}{360437}a$, $\frac{2092}{17661413}a^{11}-\frac{2092}{17661413}a^{10}+\frac{83}{2523059}a^{9}-\frac{12552}{17661413}a^{8}+\frac{85772}{17661413}a^{7}-\frac{4184}{569723}a^{6}+\frac{79496}{2523059}a^{5}-\frac{858801}{17661413}a^{4}+\frac{3165196}{17661413}a^{3}-\frac{525092}{2523059}a^{2}+\frac{89956}{360437}a-\frac{12552}{51491}$, $\frac{5553}{2523059}a^{11}+\frac{3029}{2523059}a^{10}+\frac{5100}{360437}a^{9}+\frac{13505}{2523059}a^{8}+\frac{208080}{2523059}a^{7}-\frac{30182}{2523059}a^{6}+\frac{171861}{360437}a^{5}-\frac{248880}{2523059}a^{4}+\frac{6386361}{2523059}a^{3}+\frac{1020}{51491}a^{2}+\frac{79778}{32767}a-\frac{89849}{51491}$, $\frac{249}{2523059}a^{11}-\frac{326}{360437}a^{10}+\frac{3174}{2523059}a^{9}-\frac{21689}{2523059}a^{8}+\frac{331}{360437}a^{7}-\frac{11665}{360437}a^{6}+\frac{75184}{2523059}a^{5}-\frac{800377}{2523059}a^{4}+\frac{132779}{360437}a^{3}-\frac{3373366}{2523059}a^{2}-\frac{32072}{360437}a-\frac{98451}{51491}$ 83/2523059*a^11 + 204/360437*a^10 - 1224/2523059*a^9 + 8364/2523059*a^8 - 809/360437*a^7 + 8115/360437*a^6 - 71604/2523059*a^5 + 308652/2523059*a^4 - 51204/360437*a^3 + 2140312/2523059*a^2 - 354278/360437*a + 9996/51491, 82/360437*a^11 + 66/32767*a^6 - 228167/360437*a, 2092/17661413*a^11 - 2092/17661413*a^10 + 83/2523059*a^9 - 12552/17661413*a^8 + 85772/17661413*a^7 - 4184/569723*a^6 + 79496/2523059*a^5 - 858801/17661413*a^4 + 3165196/17661413*a^3 - 525092/2523059*a^2 + 89956/360437*a - 12552/51491, 5553/2523059*a^11 + 3029/2523059*a^10 + 5100/360437*a^9 + 13505/2523059*a^8 + 208080/2523059*a^7 - 30182/2523059*a^6 + 171861/360437*a^5 - 248880/2523059*a^4 + 6386361/2523059*a^3 + 1020/51491*a^2 + 79778/32767*a - 89849/51491, 249/2523059*a^11 - 326/360437*a^10 + 3174/2523059*a^9 - 21689/2523059*a^8 + 331/360437*a^7 - 11665/360437*a^6 + 75184/2523059*a^5 - 800377/2523059*a^4 + 132779/360437*a^3 - 3373366/2523059*a^2 - 32072/360437*a - 98451/51491 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: $$1234.55163261$$ 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^{0}\cdot(2\pi)^{6}\cdot 1234.55163261 \cdot 13}{10\cdot\sqrt{33171021564453125}}\cr\approx \mathstrut & 0.542191116043 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401)

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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401, 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);

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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 + 7*x^10 - 6*x^9 + 41*x^8 - 62*x^7 + 266*x^6 - 351*x^5 + 1513*x^4 - 1757*x^3 + 2107*x^2 - 2058*x + 2401);

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

$C_{12}$ (as 12T1):

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 cyclic group of order 12 The 12 conjugacy class representatives for $C_{12}$ Character table for $C_{12}$

## Intermediate fields

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

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]

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.1.0.1}{1} }^{12}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ R ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.1.0.1}{1} }^{12}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

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

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

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

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))]; # to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_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$LabelPolynomial$efc$Galois group Slope content $$5$$ 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3} $$19$$ 19.6.4.3x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$19.6.4.3$x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$$3$$2$$4$$C_6[\ ]_{3}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.5.4t1.a.a$1 5 $$$\Q(\zeta_{5})$$$C_4$(as 4T1)$0-1$* 1.5.2t1.a.a$1 5 $$$\Q(\sqrt{5})$$$C_2$(as 2T1)$11$* 1.5.4t1.a.b$1 5 $$$\Q(\zeta_{5})$$$C_4$(as 4T1)$0-1$* 1.19.3t1.a.a$1 19 $3.3.361.1$C_3$(as 3T1)$01$* 1.95.12t1.a.a$1 5 \cdot 19 $12.0.33171021564453125.1$C_{12}$(as 12T1)$0-1$* 1.95.6t1.a.a$1 5 \cdot 19 $6.6.16290125.1$C_6$(as 6T1)$01$* 1.95.12t1.a.b$1 5 \cdot 19 $12.0.33171021564453125.1$C_{12}$(as 12T1)$0-1$* 1.19.3t1.a.b$1 19 $3.3.361.1$C_3$(as 3T1)$01$* 1.95.12t1.a.c$1 5 \cdot 19 $12.0.33171021564453125.1$C_{12}$(as 12T1)$0-1$* 1.95.6t1.a.b$1 5 \cdot 19 $6.6.16290125.1$C_6$(as 6T1)$01$* 1.95.12t1.a.d$1 5 \cdot 19 $12.0.33171021564453125.1$C_{12}$(as 12T1)$0-1\$

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 *.