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Properties

Label 24.0.13308468433...6961.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 7^{20}\cdot 11^{12}$
Root discriminant $29.07$
Ramified primes $3, 7, 11$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, -177147, 177147, -157464, 52488, -52488, 26973, -29160, 15714, -8883, 6696, -2712, 1633, -904, 744, -329, 194, -120, 37, -24, 8, -8, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441)
 
gp: K = bnfinit(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 3 x^{22} - 8 x^{21} + 8 x^{20} - 24 x^{19} + 37 x^{18} - 120 x^{17} + 194 x^{16} - 329 x^{15} + 744 x^{14} - 904 x^{13} + 1633 x^{12} - 2712 x^{11} + 6696 x^{10} - 8883 x^{9} + 15714 x^{8} - 29160 x^{7} + 26973 x^{6} - 52488 x^{5} + 52488 x^{4} - 157464 x^{3} + 177147 x^{2} - 177147 x + 531441 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(133084684332123489494188901166116961=3^{12}\cdot 7^{20}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.07$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(1,·)$, $\chi_{231}(131,·)$, $\chi_{231}(197,·)$, $\chi_{231}(100,·)$, $\chi_{231}(65,·)$, $\chi_{231}(10,·)$, $\chi_{231}(76,·)$, $\chi_{231}(142,·)$, $\chi_{231}(208,·)$, $\chi_{231}(67,·)$, $\chi_{231}(23,·)$, $\chi_{231}(89,·)$, $\chi_{231}(155,·)$, $\chi_{231}(221,·)$, $\chi_{231}(199,·)$, $\chi_{231}(32,·)$, $\chi_{231}(34,·)$, $\chi_{231}(164,·)$, $\chi_{231}(230,·)$, $\chi_{231}(166,·)$, $\chi_{231}(43,·)$, $\chi_{231}(109,·)$, $\chi_{231}(122,·)$, $\chi_{231}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{117} a^{14} - \frac{1}{9} a^{13} - \frac{1}{3} a^{12} - \frac{2}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{2}{9} a^{8} + \frac{10}{39} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{4}{13}$, $\frac{1}{351} a^{15} - \frac{1}{351} a^{14} + \frac{1}{9} a^{13} + \frac{7}{27} a^{12} + \frac{11}{27} a^{11} + \frac{1}{9} a^{10} - \frac{2}{27} a^{9} + \frac{23}{117} a^{8} - \frac{121}{351} a^{7} + \frac{10}{27} a^{6} - \frac{4}{9} a^{5} + \frac{8}{27} a^{4} + \frac{1}{27} a^{3} - \frac{4}{9} a^{2} - \frac{17}{39} a - \frac{3}{13}$, $\frac{1}{1053} a^{16} - \frac{1}{1053} a^{15} + \frac{1}{351} a^{14} - \frac{11}{81} a^{13} + \frac{11}{81} a^{12} - \frac{11}{27} a^{11} - \frac{20}{81} a^{10} + \frac{140}{351} a^{9} + \frac{113}{1053} a^{8} + \frac{103}{1053} a^{7} - \frac{10}{27} a^{6} - \frac{28}{81} a^{5} + \frac{28}{81} a^{4} - \frac{1}{27} a^{3} + \frac{16}{39} a^{2} - \frac{16}{39} a + \frac{3}{13}$, $\frac{1}{3159} a^{17} - \frac{1}{3159} a^{16} + \frac{1}{1053} a^{15} - \frac{8}{3159} a^{14} + \frac{38}{243} a^{13} - \frac{38}{81} a^{12} + \frac{115}{243} a^{11} - \frac{445}{1053} a^{10} + \frac{1166}{3159} a^{9} - \frac{1301}{3159} a^{8} + \frac{167}{1053} a^{7} + \frac{80}{243} a^{6} - \frac{80}{243} a^{5} - \frac{1}{81} a^{4} - \frac{49}{117} a^{3} + \frac{49}{117} a^{2} - \frac{10}{39} a + \frac{6}{13}$, $\frac{1}{18954} a^{18} + \frac{1}{9477} a^{17} + \frac{1}{18954} a^{15} - \frac{8}{9477} a^{14} - \frac{227}{1458} a^{12} + \frac{526}{1053} a^{11} - \frac{2999}{9477} a^{10} - \frac{317}{1458} a^{9} + \frac{19}{39} a^{8} - \frac{4439}{9477} a^{7} - \frac{83}{1458} a^{6} - \frac{80}{1053} a^{4} + \frac{253}{702} a^{3} - \frac{2}{39} a - \frac{9}{26}$, $\frac{1}{133682562} a^{19} - \frac{638}{66841281} a^{18} + \frac{463}{7426809} a^{17} - \frac{10889}{133682562} a^{16} + \frac{15688}{66841281} a^{15} - \frac{3704}{7426809} a^{14} + \frac{1588741}{10283274} a^{13} + \frac{2135477}{7426809} a^{12} - \frac{33385187}{66841281} a^{11} - \frac{23423669}{133682562} a^{10} - \frac{570436}{7426809} a^{9} - \frac{24266540}{66841281} a^{8} + \frac{30788947}{133682562} a^{7} + \frac{266432}{571293} a^{6} - \frac{376343}{825201} a^{5} + \frac{630407}{1650402} a^{4} - \frac{78964}{275067} a^{3} + \frac{6782}{30563} a^{2} + \frac{9939}{61126} a - \frac{9026}{30563}$, $\frac{1}{401047686} a^{20} - \frac{1}{401047686} a^{19} - \frac{1747}{66841281} a^{18} - \frac{6857}{401047686} a^{17} - \frac{14113}{401047686} a^{16} + \frac{5309}{66841281} a^{15} + \frac{54829}{401047686} a^{14} + \frac{6067175}{133682562} a^{13} + \frac{68094349}{200523843} a^{12} + \frac{152359321}{401047686} a^{11} + \frac{24940817}{133682562} a^{10} + \frac{29282029}{200523843} a^{9} + \frac{12495343}{401047686} a^{8} + \frac{37707425}{133682562} a^{7} - \frac{10934744}{22280427} a^{6} + \frac{2757635}{14853618} a^{5} - \frac{920833}{4951206} a^{4} + \frac{364441}{825201} a^{3} + \frac{25547}{550134} a^{2} - \frac{16877}{61126} a - \frac{8622}{30563}$, $\frac{1}{1203143058} a^{21} - \frac{1}{1203143058} a^{20} + \frac{1}{401047686} a^{19} + \frac{305}{46274733} a^{18} - \frac{103429}{1203143058} a^{17} + \frac{39763}{401047686} a^{16} - \frac{174982}{601571529} a^{15} - \frac{550165}{401047686} a^{14} + \frac{94073027}{1203143058} a^{13} - \frac{2424130}{601571529} a^{12} + \frac{1513691}{30849822} a^{11} + \frac{582544049}{1203143058} a^{10} - \frac{46722778}{601571529} a^{9} - \frac{26025637}{401047686} a^{8} - \frac{4645441}{44560854} a^{7} + \frac{3508952}{22280427} a^{6} - \frac{2444959}{14853618} a^{5} + \frac{61487}{126954} a^{4} - \frac{150613}{825201} a^{3} + \frac{125191}{550134} a^{2} + \frac{79927}{183378} a + \frac{3817}{61126}$, $\frac{1}{3609429174} a^{22} - \frac{1}{3609429174} a^{21} + \frac{1}{1203143058} a^{20} - \frac{4}{1804714587} a^{19} - \frac{33692}{1804714587} a^{18} - \frac{175913}{1203143058} a^{17} + \frac{196484}{1804714587} a^{16} - \frac{620696}{601571529} a^{15} + \frac{6699785}{3609429174} a^{14} - \frac{51563428}{1804714587} a^{13} - \frac{71855759}{601571529} a^{12} + \frac{88239443}{3609429174} a^{11} - \frac{183324175}{1804714587} a^{10} - \frac{140792762}{601571529} a^{9} - \frac{979177}{133682562} a^{8} - \frac{25134913}{66841281} a^{7} - \frac{2781121}{22280427} a^{6} - \frac{62875}{550134} a^{5} - \frac{643358}{2475603} a^{4} - \frac{101704}{275067} a^{3} + \frac{173989}{550134} a^{2} + \frac{67975}{183378} a - \frac{17727}{61126}$, $\frac{1}{10828287522} a^{23} - \frac{1}{10828287522} a^{22} + \frac{1}{3609429174} a^{21} - \frac{4}{5414143761} a^{20} + \frac{4}{5414143761} a^{19} - \frac{42448}{1804714587} a^{18} + \frac{39545}{416472597} a^{17} - \frac{669242}{1804714587} a^{16} + \frac{3417892}{5414143761} a^{15} - \frac{7829503}{5414143761} a^{14} - \frac{133470299}{1804714587} a^{13} + \frac{2474930734}{5414143761} a^{12} - \frac{1780524457}{5414143761} a^{11} + \frac{2108956}{138824199} a^{10} + \frac{55081483}{200523843} a^{9} - \frac{74558275}{200523843} a^{8} - \frac{11001596}{66841281} a^{7} + \frac{3397267}{22280427} a^{6} - \frac{2227603}{7426809} a^{5} + \frac{447100}{2475603} a^{4} + \frac{1373}{7053} a^{3} + \frac{128431}{550134} a^{2} + \frac{57497}{183378} a - \frac{3427}{61126}$

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

Class group and class number

$C_{9}$, which has order $9$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2573}{3609429174} a^{23} - \frac{2573}{1203143058} a^{22} + \frac{10292}{1804714587} a^{21} - \frac{10292}{1804714587} a^{20} + \frac{10292}{601571529} a^{19} - \frac{95201}{3609429174} a^{18} + \frac{81499}{1804714587} a^{17} - \frac{249581}{1804714587} a^{16} + \frac{846517}{3609429174} a^{15} - \frac{319052}{601571529} a^{14} + \frac{1162996}{1804714587} a^{13} - \frac{4201709}{3609429174} a^{12} + \frac{1162996}{601571529} a^{11} - \frac{12848555}{1804714587} a^{10} + \frac{846517}{133682562} a^{9} - \frac{249581}{22280427} a^{8} + \frac{51460}{2475603} a^{7} - \frac{95201}{4951206} a^{6} + \frac{10292}{275067} a^{5} - \frac{10292}{275067} a^{4} + \frac{13051}{183378} a^{3} - \frac{7719}{61126} a^{2} + \frac{7719}{61126} a - \frac{23157}{61126} \) (order $42$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 55738390.68907589 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{77})\), \(\Q(\sqrt{21}, \sqrt{33})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{-7}, \sqrt{33})\), \(\Q(\sqrt{-11}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.6.22370117.1, 6.0.603993159.1, 6.0.64827.1, 6.6.86284737.1, \(\Q(\zeta_{21})^+\), \(\Q(\zeta_{7})\), 6.0.3195731.1, 8.0.2847396321.1, 12.0.364807736118799281.1, 12.12.364807736118799281.1, 12.0.500422134593689.1, 12.0.364807736118799281.3, 12.0.364807736118799281.2, 12.0.7445055839159169.1, \(\Q(\zeta_{21})\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$