Properties

Label 12.0.7402760456175616.2
Degree 1212
Signature [0,6][0, 6]
Discriminant 7.403×10157.403\times 10^{15}
Root discriminant 21.0121.01
Ramified primes 2,7,192,7,19
Class number 33
Class group [3]
Galois group C6×S3C_6\times S_3 (as 12T18)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 1)
 
gp: K = bnfinit(y^12 + y^10 + 10*y^8 + 33*y^6 + 50*y^4 + 5*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 1)
 

x12+x10+10x8+33x6+50x4+5x2+1 x^{12} + x^{10} + 10x^{8} + 33x^{6} + 50x^{4} + 5x^{2} + 1 Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  1212
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [0,6][0, 6]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   74027604561756167402760456175616 =21674196\medspace = 2^{16}\cdot 7^{4}\cdot 19^{6} Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  21.0121.01
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:  24/372/3191/240.1928517654302842^{4/3}7^{2/3}19^{1/2}\approx 40.192851765430284
Ramified primes:   22, 77, 1919 Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q\Q
Aut(K/Q)\Aut(K/\Q):   C6C_6
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, 12a612a512a312a12\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}, 12a712a512a412a312a212\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}, 12a812a412\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}, 12a912a512a\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a, 1638a10147638a8+37319a612a573638a412a3311638a212a111319\frac{1}{638}a^{10}-\frac{147}{638}a^{8}+\frac{37}{319}a^{6}-\frac{1}{2}a^{5}-\frac{73}{638}a^{4}-\frac{1}{2}a^{3}-\frac{311}{638}a^{2}-\frac{1}{2}a-\frac{111}{319}, 1638a11147638a9+37319a7+123319a512a4+4319a312a2+97638a12\frac{1}{638}a^{11}-\frac{147}{638}a^{9}+\frac{37}{319}a^{7}+\frac{123}{319}a^{5}-\frac{1}{2}a^{4}+\frac{4}{319}a^{3}-\frac{1}{2}a^{2}+\frac{97}{638}a-\frac{1}{2} Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

C3C_{3}, which has order 33

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:  55
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   70319a11+155638a9+714319a7+4773638a5+3750319a3+1777638a \frac{70}{319} a^{11} + \frac{155}{638} a^{9} + \frac{714}{319} a^{7} + \frac{4773}{638} a^{5} + \frac{3750}{319} a^{3} + \frac{1777}{638} a  (order 44) Copy content Toggle raw display
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:   75638a11+70319a9+765638a7+1569319a5+5385638a3+1245319a\frac{75}{638}a^{11}+\frac{70}{319}a^{9}+\frac{765}{638}a^{7}+\frac{1569}{319}a^{5}+\frac{5385}{638}a^{3}+\frac{1245}{319}a, aa, 677638a11+149638a10+647638a9+54319a8+3357319a7+728319a6+22035638a5+4435638a4+32531638a3+3148319a2+775319a589319\frac{677}{638}a^{11}+\frac{149}{638}a^{10}+\frac{647}{638}a^{9}+\frac{54}{319}a^{8}+\frac{3357}{319}a^{7}+\frac{728}{319}a^{6}+\frac{22035}{638}a^{5}+\frac{4435}{638}a^{4}+\frac{32531}{638}a^{3}+\frac{3148}{319}a^{2}+\frac{775}{319}a-\frac{589}{319}, 185638a1157319a10+239638a9149638a8+1887638a71099638a6+3296319a54119638a4+5525319a33008319a2+1157319a106319\frac{185}{638}a^{11}-\frac{57}{319}a^{10}+\frac{239}{638}a^{9}-\frac{149}{638}a^{8}+\frac{1887}{638}a^{7}-\frac{1099}{638}a^{6}+\frac{3296}{319}a^{5}-\frac{4119}{638}a^{4}+\frac{5525}{319}a^{3}-\frac{3008}{319}a^{2}+\frac{1157}{319}a-\frac{106}{319}, 313638a11+13319a10+122319a9+3319a8+3065638a7+329638a6+9689638a5+327319a4+6675319a3+742319a2901638a+927638\frac{313}{638}a^{11}+\frac{13}{319}a^{10}+\frac{122}{319}a^{9}+\frac{3}{319}a^{8}+\frac{3065}{638}a^{7}+\frac{329}{638}a^{6}+\frac{9689}{638}a^{5}+\frac{327}{319}a^{4}+\frac{6675}{319}a^{3}+\frac{742}{319}a^{2}-\frac{901}{638}a+\frac{927}{638} Copy content Toggle raw display
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:  583.8867703020712 583.8867703020712
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(20(2π)6583.8867703020712347402760456175616(0.313164309242138 \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 583.8867703020712 \cdot 3}{4\cdot\sqrt{7402760456175616}}\cr\approx \mathstrut & 0.313164309242138 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + x^10 + 10*x^8 + 33*x^6 + 50*x^4 + 5*x^2 + 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

C6×S3C_6\times S_3 (as 12T18):

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 36
The 18 conjugacy class representatives for C6×S3C_6\times S_3
Character table for C6×S3C_6\times S_3

Intermediate fields

Q(19)\Q(\sqrt{19}) , Q(19)\Q(\sqrt{-19}) , Q(1)\Q(\sqrt{-1}) , Q(i,19)\Q(i, \sqrt{19}), 6.0.5377456.1

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]
 

Sibling fields

Galois closure: deg 36
Degree 18 siblings: 18.6.74933977343871482567342424064.2, deg 18
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R 62{\href{/padicField/3.6.0.1}{6} }^{2} 34{\href{/padicField/5.3.0.1}{3} }^{4} R 6,23{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3} 26{\href{/padicField/13.2.0.1}{2} }^{6} 32,16{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6} R 6,23{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3} 26{\href{/padicField/29.2.0.1}{2} }^{6} 62{\href{/padicField/31.6.0.1}{6} }^{2} 62{\href{/padicField/37.6.0.1}{6} }^{2} 26{\href{/padicField/41.2.0.1}{2} }^{6} 26{\href{/padicField/43.2.0.1}{2} }^{6} 6,23{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3} 62{\href{/padicField/53.6.0.1}{6} }^{2} 62{\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.2.6.16a2.1x12+6x11+21x10+50x9+90x8+128x7+149x6+144x5+116x4+76x3+41x2+18x+5x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 128 x^{7} + 149 x^{6} + 144 x^{5} + 116 x^{4} + 76 x^{3} + 41 x^{2} + 18 x + 566221616C6×S3C_6\times S_3[2]36[2]_{3}^{6}
77 Copy content Toggle raw display 7.2.3.4a1.3x6+18x5+117x4+324x3+351x2+169x+55x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 55332244C6C_6[ ]32[\ ]_{3}^{2}
7.6.1.0a1.1x6+x4+5x3+4x2+6x+3x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3116600C6C_6[ ]6[\ ]^{6}
1919 Copy content Toggle raw display 19.6.2.6a1.2x12+34x9+34x8+12x7+293x6+578x5+493x4+272x3+104x2+24x+23x^{12} + 34 x^{9} + 34 x^{8} + 12 x^{7} + 293 x^{6} + 578 x^{5} + 493 x^{4} + 272 x^{3} + 104 x^{2} + 24 x + 23226666C6×C2C_6\times C_2[ ]26[\ ]_{2}^{6}

Artin representations

Label Dimension Conductor Artin stem field GG Ind χ(c)\chi(c)
* 1.1.1t1.a.a11 11 Q\Q C1C_1 11 11
* 1.4.2t1.a.a11 22 2^{2} Q(1)\Q(\sqrt{-1}) C2C_2 (as 2T1) 11 1-1
* 1.19.2t1.a.a11 19 19 Q(19)\Q(\sqrt{-19}) C2C_2 (as 2T1) 11 1-1
* 1.76.2t1.a.a11 2219 2^{2} \cdot 19 Q(19)\Q(\sqrt{19}) C2C_2 (as 2T1) 11 11
1.7.3t1.a.a11 7 7 Q(ζ7)+\Q(\zeta_{7})^+ C3C_3 (as 3T1) 00 11
1.7.3t1.a.b11 7 7 Q(ζ7)+\Q(\zeta_{7})^+ C3C_3 (as 3T1) 00 11
1.133.6t1.i.a11 719 7 \cdot 19 6.0.16468459.1 C6C_6 (as 6T1) 00 1-1
1.133.6t1.i.b11 719 7 \cdot 19 6.0.16468459.1 C6C_6 (as 6T1) 00 1-1
1.532.6t1.h.a11 22719 2^{2} \cdot 7 \cdot 19 6.6.1053981376.1 C6C_6 (as 6T1) 00 11
1.28.6t1.a.a11 227 2^{2} \cdot 7 6.0.153664.1 C6C_6 (as 6T1) 00 1-1
1.28.6t1.a.b11 227 2^{2} \cdot 7 6.0.153664.1 C6C_6 (as 6T1) 00 1-1
1.532.6t1.h.b11 22719 2^{2} \cdot 7 \cdot 19 6.6.1053981376.1 C6C_6 (as 6T1) 00 11
2.3724.3t2.b.a22 227219 2^{2} \cdot 7^{2} \cdot 19 3.1.3724.1 S3S_3 (as 3T2) 11 00
2.14896.6t3.k.a22 247219 2^{4} \cdot 7^{2} \cdot 19 6.0.221890816.1 D6D_{6} (as 6T3) 11 00
* 2.2128.12t18.b.a22 24719 2^{4} \cdot 7 \cdot 19 12.0.7402760456175616.2 C6×S3C_6\times S_3 (as 12T18) 00 00
* 2.532.6t5.c.a22 22719 2^{2} \cdot 7 \cdot 19 6.0.5377456.1 S3×C3S_3\times C_3 (as 6T5) 00 00
* 2.2128.12t18.b.b22 24719 2^{4} \cdot 7 \cdot 19 12.0.7402760456175616.2 C6×S3C_6\times S_3 (as 12T18) 00 00
* 2.532.6t5.c.b22 22719 2^{2} \cdot 7 \cdot 19 6.0.5377456.1 S3×C3S_3\times C_3 (as 6T5) 00 00

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)