Properties

Label 24.0.180...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.808\times 10^{31}$
Root discriminant $20.06$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{12}\times S_3$ (as 24T65)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241)
 
gp: K = bnfinit(x^24 - 6*x^23 + 23*x^22 - 58*x^21 + 100*x^20 - 160*x^19 + 242*x^18 - 156*x^17 - 228*x^16 + 452*x^15 + 164*x^14 - 1190*x^13 + 1288*x^12 - 326*x^11 - 800*x^10 + 832*x^9 + 514*x^8 - 1302*x^7 + 500*x^6 + 664*x^5 - 155*x^4 - 1418*x^3 + 1951*x^2 - 1112*x + 241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![241, -1112, 1951, -1418, -155, 664, 500, -1302, 514, 832, -800, -326, 1288, -1190, 164, 452, -228, -156, 242, -160, 100, -58, 23, -6, 1]);
 

\( x^{24} - 6 x^{23} + 23 x^{22} - 58 x^{21} + 100 x^{20} - 160 x^{19} + 242 x^{18} - 156 x^{17} - 228 x^{16} + 452 x^{15} + 164 x^{14} - 1190 x^{13} + 1288 x^{12} - 326 x^{11} - 800 x^{10} + 832 x^{9} + 514 x^{8} - 1302 x^{7} + 500 x^{6} + 664 x^{5} - 155 x^{4} - 1418 x^{3} + 1951 x^{2} - 1112 x + 241 \)

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

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(18075490334784000000000000000000\)\(\medspace = 2^{24}\cdot 3^{24}\cdot 5^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.06$
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:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $12$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19} a^{20} - \frac{4}{19} a^{19} - \frac{7}{19} a^{18} - \frac{7}{19} a^{17} + \frac{7}{19} a^{16} + \frac{6}{19} a^{15} + \frac{9}{19} a^{14} + \frac{1}{19} a^{13} - \frac{6}{19} a^{12} + \frac{9}{19} a^{11} + \frac{7}{19} a^{10} - \frac{3}{19} a^{9} + \frac{2}{19} a^{8} + \frac{2}{19} a^{7} - \frac{6}{19} a^{6} - \frac{5}{19} a^{5} - \frac{3}{19} a^{4} + \frac{9}{19} a^{2} + \frac{1}{19} a - \frac{3}{19}$, $\frac{1}{19} a^{21} - \frac{4}{19} a^{19} + \frac{3}{19} a^{18} - \frac{2}{19} a^{17} - \frac{4}{19} a^{16} - \frac{5}{19} a^{15} - \frac{1}{19} a^{14} - \frac{2}{19} a^{13} + \frac{4}{19} a^{12} + \frac{5}{19} a^{11} + \frac{6}{19} a^{10} + \frac{9}{19} a^{9} - \frac{9}{19} a^{8} + \frac{2}{19} a^{7} + \frac{9}{19} a^{6} - \frac{4}{19} a^{5} + \frac{7}{19} a^{4} + \frac{9}{19} a^{3} - \frac{1}{19} a^{2} + \frac{1}{19} a + \frac{7}{19}$, $\frac{1}{520391} a^{22} + \frac{4421}{520391} a^{21} - \frac{652}{520391} a^{20} - \frac{229390}{520391} a^{19} + \frac{251383}{520391} a^{18} - \frac{42842}{520391} a^{17} + \frac{253826}{520391} a^{16} + \frac{244452}{520391} a^{15} - \frac{93874}{520391} a^{14} + \frac{241295}{520391} a^{13} - \frac{55259}{520391} a^{12} - \frac{117101}{520391} a^{11} - \frac{253938}{520391} a^{10} + \frac{4691}{27389} a^{9} - \frac{143170}{520391} a^{8} + \frac{79622}{520391} a^{7} + \frac{228524}{520391} a^{6} - \frac{3747}{8531} a^{5} + \frac{228049}{520391} a^{4} + \frac{5531}{520391} a^{3} - \frac{106620}{520391} a^{2} - \frac{195112}{520391} a - \frac{231627}{520391}$, $\frac{1}{419956795432079825530364746485222889} a^{23} - \frac{101319215942595358359036661784}{419956795432079825530364746485222889} a^{22} - \frac{10748810243223182880580531160791426}{419956795432079825530364746485222889} a^{21} + \frac{8573334410105795729091173607438655}{419956795432079825530364746485222889} a^{20} + \frac{2835713908526824366363550393916869}{22102989233267359238440249815011731} a^{19} - \frac{186833622028223032850129027066324130}{419956795432079825530364746485222889} a^{18} + \frac{103163242683464533647967380650929371}{419956795432079825530364746485222889} a^{17} + \frac{25003593580737856436994456804448162}{419956795432079825530364746485222889} a^{16} + \frac{23820997075217078695098363725759847}{419956795432079825530364746485222889} a^{15} + \frac{9897372513393668559421471034012683}{419956795432079825530364746485222889} a^{14} - \frac{16664015453315454594600495449082397}{419956795432079825530364746485222889} a^{13} - \frac{32223800907723303119921687129274460}{419956795432079825530364746485222889} a^{12} + \frac{146723562476124758328321705184496754}{419956795432079825530364746485222889} a^{11} - \frac{569496638052847335137654019400330}{14481268808002752604495336085697341} a^{10} - \frac{85508343340696381121036966465498166}{419956795432079825530364746485222889} a^{9} - \frac{154563330902656688521538083163387636}{419956795432079825530364746485222889} a^{8} + \frac{30953999633935022841384750801846882}{419956795432079825530364746485222889} a^{7} - \frac{140061901963098996446282679986627948}{419956795432079825530364746485222889} a^{6} - \frac{21131501790739865450536250494917081}{419956795432079825530364746485222889} a^{5} - \frac{172380349097524661189657697362642735}{419956795432079825530364746485222889} a^{4} + \frac{106564340816968229368280771930135870}{419956795432079825530364746485222889} a^{3} + \frac{109633136719095248228510667763892122}{419956795432079825530364746485222889} a^{2} - \frac{113124710509798825189788672028592445}{419956795432079825530364746485222889} a + \frac{22916780383218740968384709922429}{91713648270818918001826762717891}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{73616946842978946}{385899254913200159} a^{23} - \frac{7462656356153571980}{7332085843350803021} a^{22} + \frac{27199219556983244939}{7332085843350803021} a^{21} - \frac{63003113424945170569}{7332085843350803021} a^{20} + \frac{97853678341237425469}{7332085843350803021} a^{19} - \frac{158509176010303132348}{7332085843350803021} a^{18} + \frac{232899148742777208807}{7332085843350803021} a^{17} - \frac{62853260300608451637}{7332085843350803021} a^{16} - \frac{361334797096169948987}{7332085843350803021} a^{15} + \frac{390773395557468131143}{7332085843350803021} a^{14} + \frac{491465161484057999336}{7332085843350803021} a^{13} - \frac{1336747531522870070326}{7332085843350803021} a^{12} + \frac{907671804585560377208}{7332085843350803021} a^{11} + \frac{129018987937036181}{6326217293659019} a^{10} - \frac{1014351005174727814448}{7332085843350803021} a^{9} + \frac{485370744649705117064}{7332085843350803021} a^{8} + \frac{1045162766334852803796}{7332085843350803021} a^{7} - \frac{1122011553221285078722}{7332085843350803021} a^{6} - \frac{55062761327452545061}{7332085843350803021} a^{5} + \frac{886012307751874158164}{7332085843350803021} a^{4} + \frac{19877686242970701284}{385899254913200159} a^{3} - \frac{1734124202687164364017}{7332085843350803021} a^{2} + \frac{82737670931200852412}{385899254913200159} a - \frac{508948454126450352214}{7332085843350803021} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 535957.6069656424 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 535957.6069656424 \cdot 2}{4\sqrt{18075490334784000000000000000000}}\approx 0.238623834579320$ (assuming GRH)

Galois group

$C_{12}\times S_3$ (as 24T65):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 72
The 36 conjugacy class representatives for $C_{12}\times S_3$
Character table for $C_{12}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{15})^+\), 4.0.18000.1, \(\Q(i, \sqrt{5})\), 6.0.648000.1, 8.0.324000000.1, 12.0.419904000000.1

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

Sibling fields

Degree 36 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
2Data not computed
$3$3.12.18.61$x^{12} - 12 x^{10} - 27 x^{9} + 18 x^{8} - 36 x^{7} - 33 x^{6} - 27 x^{5} + 27 x^{4} + 27 x^{3} + 27 x + 18$$6$$2$$18$$C_{12}$$[2]_{2}^{2}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$

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) $1$ $-1$
* 1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.180.6t1.b.a$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.52488000.1 $C_6$ (as 6T1) $0$ $-1$
1.180.6t1.b.b$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.52488000.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.36.6t1.b.a$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.36.6t1.b.b$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
* 1.15.4t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.60.4t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 5 $ 4.0.18000.1 $C_4$ (as 4T1) $0$ $-1$
* 1.60.4t1.a.b$1$ $ 2^{2} \cdot 3 \cdot 5 $ 4.0.18000.1 $C_4$ (as 4T1) $0$ $-1$
* 1.15.4t1.a.b$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
1.180.12t1.a.a$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.3099363912000000000.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.b.a$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.b.b$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.180.12t1.a.b$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.3099363912000000000.1 $C_{12}$ (as 12T1) $0$ $-1$
1.180.12t1.a.c$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.3099363912000000000.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.b.c$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.180.12t1.a.d$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.3099363912000000000.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.b.d$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
2.1620.3t2.b.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ 3.1.1620.1 $S_3$ (as 3T2) $1$ $0$
2.1620.6t3.g.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ 6.2.13122000.3 $D_{6}$ (as 6T3) $1$ $0$
* 2.180.12t18.a.a$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.419904000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.180.12t18.a.b$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.419904000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.180.6t5.b.a$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.648000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.180.6t5.b.b$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.648000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.8100.12t11.a.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ 12.4.193710244500000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.8100.12t11.a.b$2$ $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ 12.4.193710244500000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.2700.24t65.a.a$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ 24.0.18075490334784000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2700.24t65.a.b$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ 24.0.18075490334784000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2700.24t65.a.c$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ 24.0.18075490334784000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2700.24t65.a.d$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2}$ 24.0.18075490334784000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$

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 *.