LMFDB - Number field 24.0.987952545632990645956882874687477121.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441)

gp: K = bnfinit(y^24 - y^23 - 3*y^22 + 10*y^21 - 10*y^20 - 30*y^19 + 127*y^18 + 210*y^17 - 718*y^16 + 529*y^15 + 1830*y^14 - 7990*y^13 + 8719*y^12 + 23970*y^11 + 16470*y^10 - 14283*y^9 - 58158*y^8 - 51030*y^7 + 92583*y^6 + 65610*y^5 - 65610*y^4 - 196830*y^3 - 177147*y^2 + 177147*y + 531441, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441)

$ x^{24} - x^{23} - 3 x^{22} + 10 x^{21} - 10 x^{20} - 30 x^{19} + 127 x^{18} + 210 x^{17} - 718 x^{16} + \cdots + 531441 $ x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$987952545632990645956882874687477121$ 987952545632990645956882874687477121 $\medspace = 3^{12}\cdot 7^{20}\cdot 13^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$31.61$	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:	$3^{1/2}7^{5/6}13^{1/2}\approx 31.606810323667133$
Ramified primes:	$3$, $7$, $13$ 3, 7, 13	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$24$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$273=3\cdot 7\cdot 13$
Dirichlet character group:	$\lbrace$$\chi_{273}(64,·)$, $\chi_{273}(1,·)$, $\chi_{273}(194,·)$, $\chi_{273}(131,·)$, $\chi_{273}(142,·)$, $\chi_{273}(79,·)$, $\chi_{273}(272,·)$, $\chi_{273}(209,·)$, $\chi_{273}(25,·)$, $\chi_{273}(155,·)$, $\chi_{273}(92,·)$, $\chi_{273}(157,·)$, $\chi_{273}(38,·)$, $\chi_{273}(103,·)$, $\chi_{273}(40,·)$, $\chi_{273}(220,·)$, $\chi_{273}(170,·)$, $\chi_{273}(235,·)$, $\chi_{273}(116,·)$, $\chi_{273}(53,·)$, $\chi_{273}(118,·)$, $\chi_{273}(233,·)$, $\chi_{273}(248,·)$, $\chi_{273}(181,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{873}a^{14}-\frac{1}{9}a^{13}+\frac{1}{3}a^{12}+\frac{4}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{3}a^{9}+\frac{4}{9}a^{8}+\frac{137}{291}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{3}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a+\frac{48}{97}$, $\frac{1}{2619}a^{15}-\frac{1}{2619}a^{14}-\frac{1}{9}a^{13}-\frac{5}{27}a^{12}-\frac{4}{27}a^{11}-\frac{1}{9}a^{10}+\frac{4}{27}a^{9}+\frac{331}{873}a^{8}+\frac{74}{2619}a^{7}+\frac{1}{27}a^{6}-\frac{2}{9}a^{5}-\frac{1}{27}a^{4}+\frac{10}{27}a^{3}-\frac{2}{9}a^{2}+\frac{145}{291}a-\frac{16}{97}$, $\frac{1}{7857}a^{16}-\frac{1}{7857}a^{15}-\frac{1}{2619}a^{14}+\frac{4}{81}a^{13}-\frac{4}{81}a^{12}-\frac{4}{27}a^{11}-\frac{14}{81}a^{10}-\frac{542}{2619}a^{9}-\frac{799}{7857}a^{8}-\frac{2009}{7857}a^{7}+\frac{1}{27}a^{6}-\frac{37}{81}a^{5}+\frac{37}{81}a^{4}+\frac{10}{27}a^{3}+\frac{113}{291}a^{2}-\frac{113}{291}a-\frac{16}{97}$, $\frac{1}{23571}a^{17}-\frac{1}{23571}a^{16}-\frac{1}{7857}a^{15}+\frac{10}{23571}a^{14}-\frac{31}{243}a^{13}-\frac{4}{81}a^{12}+\frac{94}{243}a^{11}-\frac{1415}{7857}a^{10}-\frac{8656}{23571}a^{9}-\frac{7247}{23571}a^{8}-\frac{1928}{7857}a^{7}+\frac{98}{243}a^{6}-\frac{17}{243}a^{5}+\frac{37}{81}a^{4}+\frac{16}{873}a^{3}+\frac{275}{873}a^{2}-\frac{113}{291}a+\frac{7}{97}$, $\frac{1}{282852}a^{18}-\frac{1}{70713}a^{17}+\frac{19}{282852}a^{15}-\frac{10}{70713}a^{14}-\frac{599}{2916}a^{12}-\frac{1969}{7857}a^{11}-\frac{12730}{70713}a^{10}-\frac{1265}{2916}a^{9}+\frac{73}{291}a^{8}+\frac{14126}{70713}a^{7}+\frac{337}{2916}a^{6}-\frac{1540}{7857}a^{4}-\frac{2683}{10476}a^{3}+\frac{127}{291}a-\frac{171}{388}$, $\frac{1}{24556362084}a^{19}+\frac{20737}{12278181042}a^{18}+\frac{3778}{2046363507}a^{17}-\frac{211229}{24556362084}a^{16}+\frac{77149}{12278181042}a^{15}+\frac{37780}{2046363507}a^{14}+\frac{19117345}{253158372}a^{13}-\frac{110026013}{4092727014}a^{12}+\frac{856123844}{6139090521}a^{11}-\frac{3649872857}{24556362084}a^{10}-\frac{693609497}{4092727014}a^{9}+\frac{2604919289}{6139090521}a^{8}-\frac{10986764099}{24556362084}a^{7}-\frac{636593}{42193062}a^{6}-\frac{179420500}{682121169}a^{5}+\frac{45534797}{101054988}a^{4}+\frac{5068187}{151582482}a^{3}-\frac{12357083}{25263747}a^{2}+\frac{2508313}{33684996}a-\frac{409913}{5614166}$, $\frac{1}{73669086252}a^{20}-\frac{1}{73669086252}a^{19}-\frac{11735}{12278181042}a^{18}-\frac{65609}{73669086252}a^{17}+\frac{347237}{73669086252}a^{16}-\frac{49337}{12278181042}a^{15}-\frac{656063}{73669086252}a^{14}-\frac{3651484547}{24556362084}a^{13}-\frac{10852182647}{36834543126}a^{12}-\frac{11564369309}{73669086252}a^{11}-\frac{10979535173}{24556362084}a^{10}-\frac{17332551431}{36834543126}a^{9}+\frac{32389359649}{73669086252}a^{8}-\frac{11934537827}{24556362084}a^{7}+\frac{1052974697}{4092727014}a^{6}+\frac{122269745}{2728484676}a^{5}-\frac{132705539}{909494892}a^{4}-\frac{40340921}{151582482}a^{3}-\frac{33479675}{101054988}a^{2}-\frac{14997095}{33684996}a+\frac{1601317}{5614166}$, $\frac{1}{221007258756}a^{21}-\frac{1}{221007258756}a^{20}-\frac{1}{73669086252}a^{19}+\frac{19537}{55251814689}a^{18}+\frac{1273337}{221007258756}a^{17}-\frac{528643}{73669086252}a^{16}-\frac{818237}{55251814689}a^{15}-\frac{19794215}{73669086252}a^{14}-\frac{27284847559}{221007258756}a^{13}-\frac{21598608686}{55251814689}a^{12}+\frac{17889859087}{73669086252}a^{11}+\frac{78964461875}{221007258756}a^{10}-\frac{21563797091}{55251814689}a^{9}+\frac{15311403025}{73669086252}a^{8}-\frac{3542596813}{8185454028}a^{7}+\frac{756298423}{2046363507}a^{6}-\frac{325621601}{909494892}a^{5}-\frac{85196443}{909494892}a^{4}+\frac{1592387}{75791241}a^{3}-\frac{523479}{11228332}a^{2}-\frac{3241199}{33684996}a-\frac{4349529}{11228332}$, $\frac{1}{663021776268}a^{22}-\frac{1}{663021776268}a^{21}-\frac{1}{221007258756}a^{20}+\frac{5}{331510888134}a^{19}+\frac{67463}{663021776268}a^{18}+\frac{3181049}{221007258756}a^{17}-\frac{4906471}{331510888134}a^{16}-\frac{33424445}{221007258756}a^{15}+\frac{167547377}{663021776268}a^{14}+\frac{36513914975}{331510888134}a^{13}-\frac{72483773303}{221007258756}a^{12}+\frac{180484800887}{663021776268}a^{11}-\frac{164244342223}{331510888134}a^{10}+\frac{95336793475}{221007258756}a^{9}+\frac{10509450667}{24556362084}a^{8}-\frac{1101764693}{12278181042}a^{7}-\frac{53335757}{8185454028}a^{6}-\frac{193158361}{909494892}a^{5}-\frac{39665603}{454747446}a^{4}+\frac{112710803}{303164964}a^{3}+\frac{4475901}{11228332}a^{2}-\frac{1063677}{11228332}a+\frac{2249037}{5614166}$, $\frac{1}{1989065328804}a^{23}-\frac{1}{1989065328804}a^{22}-\frac{1}{663021776268}a^{21}+\frac{5}{994532664402}a^{20}-\frac{5}{994532664402}a^{19}+\frac{490369}{331510888134}a^{18}+\frac{12694505}{994532664402}a^{17}-\frac{18212329}{331510888134}a^{16}+\frac{8272333}{994532664402}a^{15}+\frac{235119167}{994532664402}a^{14}-\frac{5878186705}{331510888134}a^{13}+\frac{253027656847}{994532664402}a^{12}-\frac{1820714329}{994532664402}a^{11}-\frac{130959612271}{331510888134}a^{10}+\frac{13319435051}{36834543126}a^{9}-\frac{2825473679}{36834543126}a^{8}+\frac{5668354513}{12278181042}a^{7}-\frac{331785605}{1364242338}a^{6}+\frac{498806995}{1364242338}a^{5}+\frac{74136901}{454747446}a^{4}-\frac{20601347}{50527494}a^{3}+\frac{42618481}{101054988}a^{2}+\frac{13729625}{33684996}a+\frac{2414473}{11228332}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/3*a^13 - 1/3*a^12 + 1/3*a^10 - 1/3*a^9 + 1/3*a^7 - 1/3*a^5 + 1/3*a^4 - 1/3*a^2 + 1/3*a, 1/873*a^14 - 1/9*a^13 + 1/3*a^12 + 4/9*a^11 - 1/9*a^10 + 1/3*a^9 + 4/9*a^8 + 137/291*a^7 - 1/9*a^6 - 2/9*a^5 - 1/3*a^4 - 1/9*a^3 - 2/9*a^2 - 1/3*a + 48/97, 1/2619*a^15 - 1/2619*a^14 - 1/9*a^13 - 5/27*a^12 - 4/27*a^11 - 1/9*a^10 + 4/27*a^9 + 331/873*a^8 + 74/2619*a^7 + 1/27*a^6 - 2/9*a^5 - 1/27*a^4 + 10/27*a^3 - 2/9*a^2 + 145/291*a - 16/97, 1/7857*a^16 - 1/7857*a^15 - 1/2619*a^14 + 4/81*a^13 - 4/81*a^12 - 4/27*a^11 - 14/81*a^10 - 542/2619*a^9 - 799/7857*a^8 - 2009/7857*a^7 + 1/27*a^6 - 37/81*a^5 + 37/81*a^4 + 10/27*a^3 + 113/291*a^2 - 113/291*a - 16/97, 1/23571*a^17 - 1/23571*a^16 - 1/7857*a^15 + 10/23571*a^14 - 31/243*a^13 - 4/81*a^12 + 94/243*a^11 - 1415/7857*a^10 - 8656/23571*a^9 - 7247/23571*a^8 - 1928/7857*a^7 + 98/243*a^6 - 17/243*a^5 + 37/81*a^4 + 16/873*a^3 + 275/873*a^2 - 113/291*a + 7/97, 1/282852*a^18 - 1/70713*a^17 + 19/282852*a^15 - 10/70713*a^14 - 599/2916*a^12 - 1969/7857*a^11 - 12730/70713*a^10 - 1265/2916*a^9 + 73/291*a^8 + 14126/70713*a^7 + 337/2916*a^6 - 1540/7857*a^4 - 2683/10476*a^3 + 127/291*a - 171/388, 1/24556362084*a^19 + 20737/12278181042*a^18 + 3778/2046363507*a^17 - 211229/24556362084*a^16 + 77149/12278181042*a^15 + 37780/2046363507*a^14 + 19117345/253158372*a^13 - 110026013/4092727014*a^12 + 856123844/6139090521*a^11 - 3649872857/24556362084*a^10 - 693609497/4092727014*a^9 + 2604919289/6139090521*a^8 - 10986764099/24556362084*a^7 - 636593/42193062*a^6 - 179420500/682121169*a^5 + 45534797/101054988*a^4 + 5068187/151582482*a^3 - 12357083/25263747*a^2 + 2508313/33684996*a - 409913/5614166, 1/73669086252*a^20 - 1/73669086252*a^19 - 11735/12278181042*a^18 - 65609/73669086252*a^17 + 347237/73669086252*a^16 - 49337/12278181042*a^15 - 656063/73669086252*a^14 - 3651484547/24556362084*a^13 - 10852182647/36834543126*a^12 - 11564369309/73669086252*a^11 - 10979535173/24556362084*a^10 - 17332551431/36834543126*a^9 + 32389359649/73669086252*a^8 - 11934537827/24556362084*a^7 + 1052974697/4092727014*a^6 + 122269745/2728484676*a^5 - 132705539/909494892*a^4 - 40340921/151582482*a^3 - 33479675/101054988*a^2 - 14997095/33684996*a + 1601317/5614166, 1/221007258756*a^21 - 1/221007258756*a^20 - 1/73669086252*a^19 + 19537/55251814689*a^18 + 1273337/221007258756*a^17 - 528643/73669086252*a^16 - 818237/55251814689*a^15 - 19794215/73669086252*a^14 - 27284847559/221007258756*a^13 - 21598608686/55251814689*a^12 + 17889859087/73669086252*a^11 + 78964461875/221007258756*a^10 - 21563797091/55251814689*a^9 + 15311403025/73669086252*a^8 - 3542596813/8185454028*a^7 + 756298423/2046363507*a^6 - 325621601/909494892*a^5 - 85196443/909494892*a^4 + 1592387/75791241*a^3 - 523479/11228332*a^2 - 3241199/33684996*a - 4349529/11228332, 1/663021776268*a^22 - 1/663021776268*a^21 - 1/221007258756*a^20 + 5/331510888134*a^19 + 67463/663021776268*a^18 + 3181049/221007258756*a^17 - 4906471/331510888134*a^16 - 33424445/221007258756*a^15 + 167547377/663021776268*a^14 + 36513914975/331510888134*a^13 - 72483773303/221007258756*a^12 + 180484800887/663021776268*a^11 - 164244342223/331510888134*a^10 + 95336793475/221007258756*a^9 + 10509450667/24556362084*a^8 - 1101764693/12278181042*a^7 - 53335757/8185454028*a^6 - 193158361/909494892*a^5 - 39665603/454747446*a^4 + 112710803/303164964*a^3 + 4475901/11228332*a^2 - 1063677/11228332*a + 2249037/5614166, 1/1989065328804*a^23 - 1/1989065328804*a^22 - 1/663021776268*a^21 + 5/994532664402*a^20 - 5/994532664402*a^19 + 490369/331510888134*a^18 + 12694505/994532664402*a^17 - 18212329/331510888134*a^16 + 8272333/994532664402*a^15 + 235119167/994532664402*a^14 - 5878186705/331510888134*a^13 + 253027656847/994532664402*a^12 - 1820714329/994532664402*a^11 - 130959612271/331510888134*a^10 + 13319435051/36834543126*a^9 - 2825473679/36834543126*a^8 + 5668354513/12278181042*a^7 - 331785605/1364242338*a^6 + 498806995/1364242338*a^5 + 74136901/454747446*a^4 - 20601347/50527494*a^3 + 42618481/101054988*a^2 + 13729625/33684996*a + 2414473/11228332

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

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (assuming GRH)

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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{1}{101054988} a^{23} - \frac{59541067}{101054988} a^{2} $ (1)/(101054988)a^(23) - (59541067)/(101054988)a^(2) (order $42$)	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:	$\frac{10}{227373723}a^{23}-\frac{10}{227373723}a^{22}-\frac{10}{75791241}a^{21}-\frac{2470}{2046363507}a^{20}-\frac{100}{227373723}a^{19}-\frac{100}{75791241}a^{18}+\frac{1270}{227373723}a^{17}+\frac{700}{75791241}a^{16}-\frac{7180}{227373723}a^{15}+\frac{5290}{227373723}a^{14}-\frac{992860}{2046363507}a^{13}-\frac{79900}{227373723}a^{12}+\frac{87190}{227373723}a^{11}+\frac{79900}{75791241}a^{10}+\frac{6100}{8421249}a^{9}-\frac{5290}{8421249}a^{8}-\frac{7180}{2807083}a^{7}-\frac{415786951}{2046363507}a^{6}+\frac{11430}{2807083}a^{5}+\frac{8100}{2807083}a^{4}-\frac{8100}{2807083}a^{3}-\frac{24300}{2807083}a^{2}-\frac{21870}{2807083}a-\frac{2785213}{2807083}$, $\frac{919480}{497266332201}a^{23}-\frac{2117473}{497266332201}a^{22}+\frac{9194800}{497266332201}a^{20}-\frac{229870}{5126457033}a^{19}+\frac{1}{2807083}a^{18}+\frac{116773960}{497266332201}a^{17}+\frac{1609090}{18417271563}a^{16}-\frac{711816931}{497266332201}a^{15}+\frac{1415999200}{497266332201}a^{14}-\frac{229870}{682121169}a^{13}-\frac{77466190}{5126457033}a^{12}+\frac{17312888920}{497266332201}a^{11}-\frac{229870}{227373723}a^{10}+\frac{616741210}{18417271563}a^{9}-\frac{1249975444}{18417271563}a^{8}-\frac{4367530}{227373723}a^{7}-\frac{49881790}{682121169}a^{6}-\frac{49881790}{227373723}a^{4}+\frac{4367530}{25263747}a^{3}+\frac{3586933}{8421249}a+\frac{2117473}{2807083}$, $\frac{173383}{221007258756}a^{23}+\frac{508}{1909251}a^{16}+\frac{173383}{1909251}a^{9}-\frac{58430071}{101054988}a^{2}-1$, $\frac{1}{33684996}a^{22}+\frac{337}{253158372}a^{21}+\frac{3370}{2046363507}a^{20}+\frac{1}{11228332}a^{19}-\frac{2683}{909494892}a^{18}-\frac{337}{75791241}a^{17}+\frac{19}{11228332}a^{16}+\frac{217}{33684996}a^{15}+\frac{1}{2187}a^{14}+\frac{4788433}{8185454028}a^{13}-\frac{171}{11228332}a^{12}-\frac{229870}{227373723}a^{11}-\frac{390583}{303164964}a^{10}+\frac{243}{11228332}a^{9}+\frac{18829}{8421249}a^{8}+\frac{39746791}{253158372}a^{7}+\frac{1643182681}{8185454028}a^{6}-\frac{13860}{2807083}a^{5}-\frac{315732481}{909494892}a^{4}-\frac{134574547}{303164964}a^{3}+\frac{30861}{2807083}a^{2}$, $\frac{7}{227373723}a^{23}-\frac{49}{909494892}a^{22}+\frac{70}{227373723}a^{20}-\frac{7}{9376236}a^{19}-\frac{2359}{909494892}a^{18}+\frac{889}{227373723}a^{17}+\frac{49}{33684996}a^{16}-\frac{16513}{909494892}a^{15}+\frac{10780}{227373723}a^{14}-\frac{63}{11228332}a^{13}-\frac{2359}{9376236}a^{12}-\frac{23590}{75791241}a^{11}-\frac{189}{11228332}a^{10}+\frac{18781}{33684996}a^{9}+\frac{2359}{2807083}a^{8}-\frac{3591}{11228332}a^{7}-\frac{13671}{11228332}a^{6}-\frac{313895401}{909494892}a^{4}+\frac{32319}{11228332}a^{3}-\frac{15309}{2807083}a+\frac{11182405}{11228332}$, $\frac{7751}{24556362084}a^{21}+\frac{23}{212139}a^{14}+\frac{9938}{212139}a^{7}+\frac{50301}{11228332}$, $\frac{1338343}{994532664402}a^{23}+\frac{162526}{497266332201}a^{22}-\frac{2028031}{221007258756}a^{21}+\frac{27180203}{1989065328804}a^{20}+\frac{24180397}{1989065328804}a^{19}-\frac{21826699}{221007258756}a^{18}+\frac{347438345}{1989065328804}a^{17}+\frac{129235519}{221007258756}a^{16}-\frac{2196816965}{1989065328804}a^{15}-\frac{1993610281}{1989065328804}a^{14}+\frac{1364896381}{221007258756}a^{13}-\frac{21199381991}{1989065328804}a^{12}-\frac{14483910379}{1989065328804}a^{11}+\frac{17138322823}{221007258756}a^{10}+\frac{1449084797}{73669086252}a^{9}-\frac{4383332419}{73669086252}a^{8}-\frac{2946813991}{24556362084}a^{7}-\frac{248464223}{2728484676}a^{6}+\frac{268919071}{909494892}a^{5}+\frac{25401443}{303164964}a^{4}-\frac{891953}{2350116}a^{3}+\frac{1560447}{11228332}a^{2}-\frac{2681789}{11228332}a+\frac{2857913}{5614166}$, $\frac{82519}{165755444067}a^{23}+\frac{970267}{221007258756}a^{22}-\frac{4875913}{663021776268}a^{21}+\frac{3176101}{663021776268}a^{20}+\frac{2956267}{221007258756}a^{19}-\frac{50715037}{663021776268}a^{18}+\frac{40968307}{663021776268}a^{17}+\frac{294074539}{663021776268}a^{16}+\frac{375969161}{663021776268}a^{15}-\frac{383113069}{221007258756}a^{14}+\frac{3180203017}{663021776268}a^{13}-\frac{4561572481}{663021776268}a^{12}-\frac{3659316883}{221007258756}a^{11}+\frac{39742042723}{663021776268}a^{10}+\frac{299562019}{24556362084}a^{9}+\frac{22954988317}{73669086252}a^{8}-\frac{3004825057}{24556362084}a^{7}-\frac{57998233}{909494892}a^{6}-\frac{2509137455}{2728484676}a^{5}+\frac{5383999}{101054988}a^{4}-\frac{93473437}{303164964}a^{3}+\frac{147891217}{101054988}a^{2}-\frac{564369}{5614166}a+\frac{3453041}{5614166}$, $\frac{13411415}{1989065328804}a^{23}-\frac{28512239}{1989065328804}a^{22}+\frac{571285}{331510888134}a^{21}+\frac{59550085}{994532664402}a^{20}-\frac{148527439}{994532664402}a^{19}+\frac{16289399}{663021776268}a^{18}+\frac{749603701}{994532664402}a^{17}+\frac{139876415}{331510888134}a^{16}-\frac{9099298589}{1989065328804}a^{15}+\frac{9737653405}{994532664402}a^{14}-\frac{820043521}{331510888134}a^{13}-\frac{93589685807}{1989065328804}a^{12}+\frac{120213051349}{994532664402}a^{11}-\frac{7185073507}{331510888134}a^{10}+\frac{43308480727}{221007258756}a^{9}-\frac{2480000401}{12278181042}a^{8}-\frac{33010793}{454747446}a^{7}-\frac{597351551}{2728484676}a^{6}+\frac{852553577}{1364242338}a^{5}-\frac{181946045}{454747446}a^{4}+\frac{20543843}{303164964}a^{3}-\frac{121056077}{101054988}a^{2}-\frac{10186721}{33684996}a+\frac{14769965}{11228332}$, $\frac{1406095}{663021776268}a^{23}-\frac{859360}{165755444067}a^{22}-\frac{518980}{55251814689}a^{21}+\frac{23552809}{663021776268}a^{20}-\frac{16104049}{663021776268}a^{19}-\frac{9845729}{110503629378}a^{18}+\frac{262250179}{663021776268}a^{17}+\frac{66915127}{221007258756}a^{16}-\frac{980123591}{331510888134}a^{15}+\frac{1119256753}{663021776268}a^{14}+\frac{1580951353}{221007258756}a^{13}-\frac{7508228393}{331510888134}a^{12}+\frac{19454728099}{663021776268}a^{11}+\frac{16281671035}{221007258756}a^{10}-\frac{2652960437}{36834543126}a^{9}-\frac{21056444375}{73669086252}a^{8}-\frac{1466955841}{8185454028}a^{7}+\frac{1284985601}{4092727014}a^{6}+\frac{3178771685}{2728484676}a^{5}+\frac{800270321}{909494892}a^{4}-\frac{102315653}{151582482}a^{3}-\frac{30532852}{25263747}a^{2}-\frac{6059361}{11228332}a-\frac{2279399}{5614166}$, $\frac{4206515}{663021776268}a^{23}-\frac{2106200}{165755444067}a^{22}-\frac{3638827}{221007258756}a^{21}+\frac{65794037}{663021776268}a^{20}-\frac{87267761}{663021776268}a^{19}-\frac{48720007}{221007258756}a^{18}+\frac{808751651}{663021776268}a^{17}+\frac{116017579}{221007258756}a^{16}-\frac{4697086031}{663021776268}a^{15}+\frac{6214168829}{663021776268}a^{14}+\frac{71409475}{5139703692}a^{13}-\frac{52042329197}{663021776268}a^{12}+\frac{71554470167}{663021776268}a^{11}+\frac{38264132275}{221007258756}a^{10}-\frac{16647028487}{73669086252}a^{9}-\frac{4675520045}{24556362084}a^{8}+\frac{1766929999}{8185454028}a^{7}-\frac{572619263}{2728484676}a^{6}+\frac{1110697409}{2728484676}a^{5}+\frac{274819529}{909494892}a^{4}-\frac{390061045}{303164964}a^{3}-\frac{7573012}{25263747}a^{2}+\frac{60263129}{33684996}a+\frac{154219}{5614166}$ 10/227373723a^23 - 10/227373723a^22 - 10/75791241a^21 - 2470/2046363507a^20 - 100/227373723a^19 - 100/75791241a^18 + 1270/227373723a^17 + 700/75791241a^16 - 7180/227373723a^15 + 5290/227373723a^14 - 992860/2046363507a^13 - 79900/227373723a^12 + 87190/227373723a^11 + 79900/75791241a^10 + 6100/8421249a^9 - 5290/8421249a^8 - 7180/2807083a^7 - 415786951/2046363507a^6 + 11430/2807083a^5 + 8100/2807083a^4 - 8100/2807083a^3 - 24300/2807083a^2 - 21870/2807083a - 2785213/2807083, 919480/497266332201a^23 - 2117473/497266332201a^22 + 9194800/497266332201a^20 - 229870/5126457033a^19 + 1/2807083a^18 + 116773960/497266332201a^17 + 1609090/18417271563a^16 - 711816931/497266332201a^15 + 1415999200/497266332201a^14 - 229870/682121169a^13 - 77466190/5126457033a^12 + 17312888920/497266332201a^11 - 229870/227373723a^10 + 616741210/18417271563a^9 - 1249975444/18417271563a^8 - 4367530/227373723a^7 - 49881790/682121169a^6 - 49881790/227373723a^4 + 4367530/25263747a^3 + 3586933/8421249a + 2117473/2807083, 173383/221007258756a^23 + 508/1909251a^16 + 173383/1909251a^9 - 58430071/101054988a^2 - 1, 1/33684996a^22 + 337/253158372a^21 + 3370/2046363507a^20 + 1/11228332a^19 - 2683/909494892a^18 - 337/75791241a^17 + 19/11228332a^16 + 217/33684996a^15 + 1/2187a^14 + 4788433/8185454028a^13 - 171/11228332a^12 - 229870/227373723a^11 - 390583/303164964a^10 + 243/11228332a^9 + 18829/8421249a^8 + 39746791/253158372a^7 + 1643182681/8185454028a^6 - 13860/2807083a^5 - 315732481/909494892a^4 - 134574547/303164964a^3 + 30861/2807083a^2, 7/227373723a^23 - 49/909494892a^22 + 70/227373723a^20 - 7/9376236a^19 - 2359/909494892a^18 + 889/227373723a^17 + 49/33684996a^16 - 16513/909494892a^15 + 10780/227373723a^14 - 63/11228332a^13 - 2359/9376236a^12 - 23590/75791241a^11 - 189/11228332a^10 + 18781/33684996a^9 + 2359/2807083a^8 - 3591/11228332a^7 - 13671/11228332a^6 - 313895401/909494892a^4 + 32319/11228332a^3 - 15309/2807083a + 11182405/11228332, 7751/24556362084a^21 + 23/212139a^14 + 9938/212139a^7 + 50301/11228332, 1338343/994532664402a^23 + 162526/497266332201a^22 - 2028031/221007258756a^21 + 27180203/1989065328804a^20 + 24180397/1989065328804a^19 - 21826699/221007258756a^18 + 347438345/1989065328804a^17 + 129235519/221007258756a^16 - 2196816965/1989065328804a^15 - 1993610281/1989065328804a^14 + 1364896381/221007258756a^13 - 21199381991/1989065328804a^12 - 14483910379/1989065328804a^11 + 17138322823/221007258756a^10 + 1449084797/73669086252a^9 - 4383332419/73669086252a^8 - 2946813991/24556362084a^7 - 248464223/2728484676a^6 + 268919071/909494892a^5 + 25401443/303164964a^4 - 891953/2350116a^3 + 1560447/11228332a^2 - 2681789/11228332a + 2857913/5614166, 82519/165755444067a^23 + 970267/221007258756a^22 - 4875913/663021776268a^21 + 3176101/663021776268a^20 + 2956267/221007258756a^19 - 50715037/663021776268a^18 + 40968307/663021776268a^17 + 294074539/663021776268a^16 + 375969161/663021776268a^15 - 383113069/221007258756a^14 + 3180203017/663021776268a^13 - 4561572481/663021776268a^12 - 3659316883/221007258756a^11 + 39742042723/663021776268a^10 + 299562019/24556362084a^9 + 22954988317/73669086252a^8 - 3004825057/24556362084a^7 - 57998233/909494892a^6 - 2509137455/2728484676a^5 + 5383999/101054988a^4 - 93473437/303164964a^3 + 147891217/101054988a^2 - 564369/5614166a + 3453041/5614166, 13411415/1989065328804a^23 - 28512239/1989065328804a^22 + 571285/331510888134a^21 + 59550085/994532664402a^20 - 148527439/994532664402a^19 + 16289399/663021776268a^18 + 749603701/994532664402a^17 + 139876415/331510888134a^16 - 9099298589/1989065328804a^15 + 9737653405/994532664402a^14 - 820043521/331510888134a^13 - 93589685807/1989065328804a^12 + 120213051349/994532664402a^11 - 7185073507/331510888134a^10 + 43308480727/221007258756a^9 - 2480000401/12278181042a^8 - 33010793/454747446a^7 - 597351551/2728484676a^6 + 852553577/1364242338a^5 - 181946045/454747446a^4 + 20543843/303164964a^3 - 121056077/101054988a^2 - 10186721/33684996a + 14769965/11228332, 1406095/663021776268a^23 - 859360/165755444067a^22 - 518980/55251814689a^21 + 23552809/663021776268a^20 - 16104049/663021776268a^19 - 9845729/110503629378a^18 + 262250179/663021776268a^17 + 66915127/221007258756a^16 - 980123591/331510888134a^15 + 1119256753/663021776268a^14 + 1580951353/221007258756a^13 - 7508228393/331510888134a^12 + 19454728099/663021776268a^11 + 16281671035/221007258756a^10 - 2652960437/36834543126a^9 - 21056444375/73669086252a^8 - 1466955841/8185454028a^7 + 1284985601/4092727014a^6 + 3178771685/2728484676a^5 + 800270321/909494892a^4 - 102315653/151582482a^3 - 30532852/25263747a^2 - 6059361/11228332a - 2279399/5614166, 4206515/663021776268a^23 - 2106200/165755444067a^22 - 3638827/221007258756a^21 + 65794037/663021776268a^20 - 87267761/663021776268a^19 - 48720007/221007258756a^18 + 808751651/663021776268a^17 + 116017579/221007258756a^16 - 4697086031/663021776268a^15 + 6214168829/663021776268a^14 + 71409475/5139703692a^13 - 52042329197/663021776268a^12 + 71554470167/663021776268a^11 + 38264132275/221007258756a^10 - 16647028487/73669086252a^9 - 4675520045/24556362084a^8 + 1766929999/8185454028a^7 - 572619263/2728484676a^6 + 1110697409/2728484676a^5 + 274819529/909494892a^4 - 390061045/303164964a^3 - 7573012/25263747a^2 + 60263129/33684996a + 154219/5614166 (assuming GRH)	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:	$ 91734319.3933256 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 91734319.3933256 \cdot 32}{42\cdot\sqrt{987952545632990645956882874687477121}}\cr\approx \mathstrut & 0.266209107288151 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441)

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^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441, 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^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441);

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^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441);

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

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

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)

An abelian group of order 24

The 24 conjugacy class representatives for $C_2^2\times C_6$

Character table for $C_2^2\times C_6$

Intermediate fields

$\Q(\sqrt{273}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-91}) $, $\Q(\sqrt{-39}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{13}) $, $\Q(\sqrt{21}) $, $\Q(\zeta_{7})^+$, $\Q(\sqrt{-3}, \sqrt{-91})$, $\Q(\sqrt{-7}, \sqrt{-39})$, $\Q(\sqrt{13}, \sqrt{21})$, $\Q(\sqrt{-3}, \sqrt{13})$, $\Q(\sqrt{-3}, \sqrt{-7})$, $\Q(\sqrt{21}, \sqrt{-39})$, $\Q(\sqrt{-7}, \sqrt{13})$, 6.6.996974433.1, 6.0.64827.1, 6.0.36924979.1, 6.0.142424919.1, $\Q(\zeta_{7})$, 6.6.5274997.1, $\Q(\zeta_{21})^+$, 8.0.5554571841.1, 12.0.993958020055671489.2, 12.0.993958020055671489.3, 12.12.993958020055671489.1, 12.0.20284857552156561.1, $\Q(\zeta_{21})$, 12.0.993958020055671489.1, 12.0.1363454074150441.1

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{4}$	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	R	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.6.0.1}{6} }^{4}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.1.0.1}{1} }^{24}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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

# 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$13$ 13	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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