LMFDB - Number field 24.0.2243572222946525052726785077149696.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096)

gp: K = bnfinit(y^24 - 4*y^23 + 8*y^22 - 12*y^21 + 24*y^20 - 56*y^19 + 104*y^18 - 152*y^17 + 224*y^16 - 376*y^15 + 608*y^14 - 848*y^13 + 1156*y^12 - 1696*y^11 + 2432*y^10 - 3008*y^9 + 3584*y^8 - 4864*y^7 + 6656*y^6 - 7168*y^5 + 6144*y^4 - 6144*y^3 + 8192*y^2 - 8192*y + 4096, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096)

$ x^{24} - 4 x^{23} + 8 x^{22} - 12 x^{21} + 24 x^{20} - 56 x^{19} + 104 x^{18} - 152 x^{17} + 224 x^{16} + \cdots + 4096 $ x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096

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:	$2243572222946525052726785077149696$ 2243572222946525052726785077149696 $\medspace = 2^{32}\cdot 3^{12}\cdot 23^{4}\cdot 37^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$24.53$	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:	$2^{4/3}3^{1/2}23^{1/2}37^{1/2}\approx 127.320617115608$
Ramified primes:	$2$, $3$, $23$, $37$ 2, 3, 23, 37	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}$, $\frac{1}{16}a^{15}+\frac{1}{4}a^{3}$, $\frac{1}{32}a^{16}+\frac{1}{8}a^{4}$, $\frac{1}{64}a^{17}+\frac{1}{16}a^{5}$, $\frac{1}{256}a^{18}-\frac{1}{64}a^{15}-\frac{1}{32}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}+\frac{1}{32}a^{11}+\frac{1}{32}a^{9}-\frac{1}{8}a^{8}+\frac{1}{16}a^{7}+\frac{1}{64}a^{6}+\frac{3}{16}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}$, $\frac{1}{512}a^{19}-\frac{1}{128}a^{16}+\frac{1}{64}a^{15}+\frac{1}{64}a^{14}+\frac{1}{64}a^{13}+\frac{1}{64}a^{12}+\frac{1}{64}a^{10}-\frac{1}{32}a^{8}-\frac{31}{128}a^{7}+\frac{7}{32}a^{6}-\frac{3}{16}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{1024}a^{20}-\frac{1}{256}a^{17}+\frac{1}{128}a^{16}+\frac{1}{128}a^{15}+\frac{1}{128}a^{14}+\frac{1}{128}a^{13}+\frac{1}{128}a^{11}-\frac{1}{16}a^{10}-\frac{1}{64}a^{9}-\frac{31}{256}a^{8}+\frac{15}{64}a^{7}-\frac{3}{32}a^{6}+\frac{3}{32}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{21}-\frac{1}{512}a^{18}+\frac{1}{256}a^{17}+\frac{1}{256}a^{16}+\frac{1}{256}a^{15}+\frac{1}{256}a^{14}+\frac{1}{256}a^{12}-\frac{1}{32}a^{11}+\frac{7}{128}a^{10}-\frac{31}{512}a^{9}+\frac{15}{128}a^{8}+\frac{5}{64}a^{7}+\frac{3}{64}a^{6}-\frac{1}{4}a^{5}-\frac{3}{8}a^{4}-\frac{7}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4096}a^{22}-\frac{1}{1024}a^{19}-\frac{1}{512}a^{18}+\frac{1}{512}a^{17}+\frac{1}{512}a^{16}+\frac{9}{512}a^{15}-\frac{1}{32}a^{14}-\frac{15}{512}a^{13}-\frac{3}{64}a^{12}-\frac{1}{256}a^{11}-\frac{31}{1024}a^{10}+\frac{7}{256}a^{9}+\frac{5}{128}a^{8}-\frac{21}{128}a^{7}-\frac{9}{64}a^{6}-\frac{1}{8}a^{5}-\frac{11}{32}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1564672}a^{23}-\frac{69}{782336}a^{22}+\frac{41}{391168}a^{21}-\frac{149}{391168}a^{20}+\frac{27}{97792}a^{19}-\frac{367}{195584}a^{18}+\frac{1059}{195584}a^{17}+\frac{1707}{195584}a^{16}+\frac{1773}{97792}a^{15}-\frac{4587}{195584}a^{14}-\frac{2817}{97792}a^{13}-\frac{1519}{97792}a^{12}-\frac{22871}{391168}a^{11}-\frac{8079}{195584}a^{10}+\frac{5117}{97792}a^{9}-\frac{5627}{48896}a^{8}+\frac{1559}{24448}a^{7}+\frac{283}{3056}a^{6}-\frac{5139}{12224}a^{5}+\frac{1249}{6112}a^{4}-\frac{777}{3056}a^{3}+\frac{73}{382}a^{2}-\frac{39}{382}a+\frac{67}{382}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6, 1/2*a^7, 1/4*a^8 - 1/2*a^5 - 1/2*a^2, 1/8*a^9 - 1/4*a^6 + 1/4*a^3, 1/8*a^10 - 1/4*a^7 + 1/4*a^4, 1/8*a^11 - 1/4*a^5 - 1/2*a^2, 1/8*a^12 - 1/4*a^6 - 1/2*a^3, 1/16*a^13 - 1/8*a^7 - 1/4*a^4, 1/16*a^14 - 1/8*a^8 - 1/4*a^5, 1/16*a^15 + 1/4*a^3, 1/32*a^16 + 1/8*a^4, 1/64*a^17 + 1/16*a^5, 1/256*a^18 - 1/64*a^15 - 1/32*a^14 - 1/32*a^13 + 1/32*a^12 + 1/32*a^11 + 1/32*a^9 - 1/8*a^8 + 1/16*a^7 + 1/64*a^6 + 3/16*a^5 - 1/8*a^4 + 3/8*a^3 - 1/2, 1/512*a^19 - 1/128*a^16 + 1/64*a^15 + 1/64*a^14 + 1/64*a^13 + 1/64*a^12 + 1/64*a^10 - 1/32*a^8 - 31/128*a^7 + 7/32*a^6 - 3/16*a^5 + 3/16*a^4 + 1/4*a^3 + 1/4*a, 1/1024*a^20 - 1/256*a^17 + 1/128*a^16 + 1/128*a^15 + 1/128*a^14 + 1/128*a^13 + 1/128*a^11 - 1/16*a^10 - 1/64*a^9 - 31/256*a^8 + 15/64*a^7 - 3/32*a^6 + 3/32*a^5 - 1/2*a^4 + 1/8*a^2 - 1/2*a, 1/2048*a^21 - 1/512*a^18 + 1/256*a^17 + 1/256*a^16 + 1/256*a^15 + 1/256*a^14 + 1/256*a^12 - 1/32*a^11 + 7/128*a^10 - 31/512*a^9 + 15/128*a^8 + 5/64*a^7 + 3/64*a^6 - 1/4*a^5 - 3/8*a^4 - 7/16*a^3 + 1/4*a^2 - 1/2*a, 1/4096*a^22 - 1/1024*a^19 - 1/512*a^18 + 1/512*a^17 + 1/512*a^16 + 9/512*a^15 - 1/32*a^14 - 15/512*a^13 - 3/64*a^12 - 1/256*a^11 - 31/1024*a^10 + 7/256*a^9 + 5/128*a^8 - 21/128*a^7 - 9/64*a^6 - 1/8*a^5 - 11/32*a^4 + 1/4*a^3 + 1/4*a^2 - 1/2*a - 1/2, 1/1564672*a^23 - 69/782336*a^22 + 41/391168*a^21 - 149/391168*a^20 + 27/97792*a^19 - 367/195584*a^18 + 1059/195584*a^17 + 1707/195584*a^16 + 1773/97792*a^15 - 4587/195584*a^14 - 2817/97792*a^13 - 1519/97792*a^12 - 22871/391168*a^11 - 8079/195584*a^10 + 5117/97792*a^9 - 5627/48896*a^8 + 1559/24448*a^7 + 283/3056*a^6 - 5139/12224*a^5 + 1249/6112*a^4 - 777/3056*a^3 + 73/382*a^2 - 39/382*a + 67/382

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{2}$, which has order $2$ (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{2121}{1564672} a^{23} - \frac{2027}{391168} a^{22} + \frac{4067}{391168} a^{21} - \frac{5081}{391168} a^{20} + \frac{3945}{195584} a^{19} - \frac{7913}{195584} a^{18} + \frac{15641}{195584} a^{17} - \frac{21823}{195584} a^{16} + \frac{1663}{12224} a^{15} - \frac{36923}{195584} a^{14} + \frac{14423}{48896} a^{13} - \frac{38211}{97792} a^{12} + \frac{195609}{391168} a^{11} - \frac{8905}{12224} a^{10} + \frac{101385}{97792} a^{9} - \frac{49319}{48896} a^{8} + \frac{16473}{24448} a^{7} - \frac{8111}{12224} a^{6} + \frac{18505}{12224} a^{5} - \frac{4509}{3056} a^{4} - \frac{451}{3056} a^{3} + \frac{887}{382} a^{2} - \frac{1751}{764} a + \frac{97}{191} $ (2121)/(1564672)a^(23) - (2027)/(391168)a^(22) + (4067)/(391168)a^(21) - (5081)/(391168)a^(20) + (3945)/(195584)a^(19) - (7913)/(195584)a^(18) + (15641)/(195584)a^(17) - (21823)/(195584)a^(16) + (1663)/(12224)a^(15) - (36923)/(195584)a^(14) + (14423)/(48896)a^(13) - (38211)/(97792)a^(12) + (195609)/(391168)a^(11) - (8905)/(12224)a^(10) + (101385)/(97792)a^(9) - (49319)/(48896)a^(8) + (16473)/(24448)a^(7) - (8111)/(12224)a^(6) + (18505)/(12224)a^(5) - (4509)/(3056)a^(4) - (451)/(3056)a^(3) + (887)/(382)a^(2) - (1751)/(764)*a + (97)/(191) (order $12$)	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:	$\frac{975}{1564672}a^{23}-\frac{881}{391168}a^{22}+\frac{1393}{391168}a^{21}-\frac{879}{391168}a^{20}-\frac{257}{195584}a^{19}-\frac{655}{195584}a^{18}+\frac{2271}{195584}a^{17}-\frac{1577}{195584}a^{16}-\frac{247}{12224}a^{15}+\frac{6243}{195584}a^{14}-\frac{2003}{48896}a^{13}+\frac{6483}{97792}a^{12}-\frac{56129}{391168}a^{11}+\frac{4537}{24448}a^{10}-\frac{21237}{97792}a^{9}+\frac{22879}{48896}a^{8}-\frac{25165}{24448}a^{7}+\frac{15573}{12224}a^{6}-\frac{13201}{12224}a^{5}+\frac{2749}{3056}a^{4}-\frac{6945}{3056}a^{3}+\frac{1269}{382}a^{2}-\frac{2133}{764}a-\frac{94}{191}$, $\frac{89}{48896}a^{23}-\frac{4093}{391168}a^{22}+\frac{9617}{391168}a^{21}-\frac{3139}{97792}a^{20}+\frac{1585}{48896}a^{19}-\frac{5953}{97792}a^{18}+\frac{3599}{24448}a^{17}-\frac{5609}{24448}a^{16}+\frac{1399}{6112}a^{15}-\frac{11495}{48896}a^{14}+\frac{18883}{48896}a^{13}-\frac{27867}{48896}a^{12}+\frac{15025}{24448}a^{11}-\frac{73337}{97792}a^{10}+\frac{123493}{97792}a^{9}-\frac{15097}{12224}a^{8}-\frac{10459}{24448}a^{7}+\frac{12725}{6112}a^{6}-\frac{154}{191}a^{5}-\frac{192}{191}a^{4}-\frac{8565}{3056}a^{3}+\frac{4489}{382}a^{2}-\frac{10325}{764}a+\frac{2299}{382}$, $\frac{105}{1564672}a^{23}+\frac{293}{391168}a^{22}-\frac{213}{97792}a^{21}+\frac{17}{391168}a^{20}+\frac{1277}{195584}a^{19}-\frac{1863}{195584}a^{18}+\frac{1561}{195584}a^{17}-\frac{2979}{195584}a^{16}+\frac{2835}{48896}a^{15}-\frac{20943}{195584}a^{14}+\frac{6913}{48896}a^{13}-\frac{17009}{97792}a^{12}+\frac{130441}{391168}a^{11}-\frac{11659}{24448}a^{10}+\frac{28269}{48896}a^{9}-\frac{9377}{12224}a^{8}+\frac{36107}{24448}a^{7}-\frac{25345}{12224}a^{6}+\frac{23855}{12224}a^{5}-\frac{4047}{3056}a^{4}+\frac{2285}{764}a^{3}-\frac{6585}{1528}a^{2}+\frac{3079}{764}a-\frac{223}{382}$, $\frac{4835}{1564672}a^{23}-\frac{3327}{195584}a^{22}+\frac{3671}{97792}a^{21}-\frac{17153}{391168}a^{20}+\frac{8015}{195584}a^{19}-\frac{18009}{195584}a^{18}+\frac{44631}{195584}a^{17}-\frac{65469}{195584}a^{16}+\frac{3843}{12224}a^{15}-\frac{64165}{195584}a^{14}+\frac{13899}{24448}a^{13}-\frac{79107}{97792}a^{12}+\frac{322483}{391168}a^{11}-\frac{108097}{97792}a^{10}+\frac{94109}{48896}a^{9}-\frac{38739}{24448}a^{8}-\frac{14923}{12224}a^{7}+\frac{40225}{12224}a^{6}-\frac{7515}{12224}a^{5}-\frac{5887}{3056}a^{4}-\frac{4015}{764}a^{3}+\frac{27643}{1528}a^{2}-\frac{3653}{191}a+\frac{3065}{382}$, $\frac{1459}{1564672}a^{23}-\frac{3643}{782336}a^{22}+\frac{4811}{391168}a^{21}-\frac{10347}{391168}a^{20}+\frac{4631}{97792}a^{19}-\frac{16697}{195584}a^{18}+\frac{29305}{195584}a^{17}-\frac{50551}{195584}a^{16}+\frac{40395}{97792}a^{15}-\frac{125089}{195584}a^{14}+\frac{89705}{97792}a^{13}-\frac{128973}{97792}a^{12}+\frac{723947}{391168}a^{11}-\frac{517497}{195584}a^{10}+\frac{352863}{97792}a^{9}-\frac{235543}{48896}a^{8}+\frac{143021}{24448}a^{7}-\frac{86321}{12224}a^{6}+\frac{96359}{12224}a^{5}-\frac{56385}{6112}a^{4}+\frac{30311}{3056}a^{3}-\frac{7591}{764}a^{2}+\frac{2691}{382}a-\frac{688}{191}$, $\frac{2095}{782336}a^{23}-\frac{4361}{391168}a^{22}+\frac{4529}{195584}a^{21}-\frac{891}{24448}a^{20}+\frac{3085}{48896}a^{19}-\frac{12123}{97792}a^{18}+\frac{21723}{97792}a^{17}-\frac{32971}{97792}a^{16}+\frac{24315}{48896}a^{15}-\frac{72753}{97792}a^{14}+\frac{53001}{48896}a^{13}-\frac{72061}{48896}a^{12}+\frac{399087}{195584}a^{11}-\frac{289023}{97792}a^{10}+\frac{197925}{48896}a^{9}-\frac{232919}{48896}a^{8}+\frac{64181}{12224}a^{7}-\frac{76423}{12224}a^{6}+\frac{11815}{1528}a^{5}-\frac{24875}{3056}a^{4}+\frac{1299}{191}a^{3}-\frac{6369}{1528}a^{2}+\frac{425}{191}a-\frac{231}{382}$, $\frac{2133}{1564672}a^{23}-\frac{1635}{782336}a^{22}+\frac{137}{97792}a^{21}+\frac{7}{391168}a^{20}+\frac{27}{3056}a^{19}-\frac{4677}{195584}a^{18}+\frac{5047}{195584}a^{17}-\frac{3249}{195584}a^{16}+\frac{3829}{97792}a^{15}-\frac{22443}{195584}a^{14}+\frac{16625}{97792}a^{13}-\frac{18621}{97792}a^{12}+\frac{109101}{391168}a^{11}-\frac{97133}{195584}a^{10}+\frac{15151}{24448}a^{9}-\frac{25545}{48896}a^{8}+\frac{21047}{24448}a^{7}-\frac{21649}{12224}a^{6}+\frac{28653}{12224}a^{5}-\frac{8737}{6112}a^{4}+\frac{1129}{1528}a^{3}-\frac{2395}{764}a^{2}+\frac{1095}{191}a-\frac{1867}{382}$, $\frac{2305}{782336}a^{23}-\frac{3189}{391168}a^{22}+\frac{3549}{391168}a^{21}+\frac{29}{6112}a^{20}-\frac{1399}{97792}a^{19}+\frac{1}{24448}a^{18}+\frac{1543}{97792}a^{17}+\frac{5001}{97792}a^{16}-\frac{4615}{24448}a^{15}+\frac{27847}{97792}a^{14}-\frac{17139}{48896}a^{13}+\frac{893}{1528}a^{12}-\frac{186543}{195584}a^{11}+\frac{118061}{97792}a^{10}-\frac{159379}{97792}a^{9}+\frac{156433}{48896}a^{8}-\frac{128883}{24448}a^{7}+\frac{17481}{3056}a^{6}-\frac{13635}{3056}a^{5}+\frac{9169}{1528}a^{4}-\frac{35673}{3056}a^{3}+\frac{22299}{1528}a^{2}-\frac{7185}{764}a+\frac{405}{382}$, $\frac{2365}{1564672}a^{23}-\frac{1307}{195584}a^{22}+\frac{6431}{391168}a^{21}-\frac{10495}{391168}a^{20}+\frac{7571}{195584}a^{19}-\frac{13421}{195584}a^{18}+\frac{26501}{195584}a^{17}-\frac{43087}{195584}a^{16}+\frac{14959}{48896}a^{15}-\frac{78911}{195584}a^{14}+\frac{14437}{24448}a^{13}-\frac{81855}{97792}a^{12}+\frac{445933}{391168}a^{11}-\frac{150583}{97792}a^{10}+\frac{212337}{97792}a^{9}-\frac{16083}{6112}a^{8}+\frac{63001}{24448}a^{7}-\frac{13881}{6112}a^{6}+\frac{33975}{12224}a^{5}-\frac{5999}{1528}a^{4}+\frac{9747}{3056}a^{3}+\frac{879}{1528}a^{2}-\frac{2829}{764}a+\frac{1453}{382}$, $\frac{1825}{1564672}a^{23}-\frac{4831}{782336}a^{22}+\frac{8357}{391168}a^{21}-\frac{18659}{391168}a^{20}+\frac{8401}{97792}a^{19}-\frac{27251}{195584}a^{18}+\frac{48651}{195584}a^{17}-\frac{86649}{195584}a^{16}+\frac{70855}{97792}a^{15}-\frac{207171}{195584}a^{14}+\frac{148529}{97792}a^{13}-\frac{214303}{97792}a^{12}+\frac{1221673}{391168}a^{11}-\frac{843577}{195584}a^{10}+\frac{578501}{97792}a^{9}-\frac{196619}{24448}a^{8}+\frac{30923}{3056}a^{7}-\frac{68547}{6112}a^{6}+\frac{148787}{12224}a^{5}-\frac{88211}{6112}a^{4}+\frac{51911}{3056}a^{3}-\frac{23483}{1528}a^{2}+\frac{6821}{764}a-\frac{729}{382}$, $\frac{139}{195584}a^{23}-\frac{4339}{782336}a^{22}+\frac{3887}{195584}a^{21}-\frac{4559}{97792}a^{20}+\frac{16691}{195584}a^{19}-\frac{14389}{97792}a^{18}+\frac{25545}{97792}a^{17}-\frac{45827}{97792}a^{16}+\frac{75137}{97792}a^{15}-\frac{1757}{1528}a^{14}+\frac{162437}{97792}a^{13}-\frac{58809}{24448}a^{12}+\frac{83721}{24448}a^{11}-\frac{938355}{195584}a^{10}+\frac{161353}{24448}a^{9}-\frac{109841}{12224}a^{8}+\frac{278379}{24448}a^{7}-\frac{80937}{6112}a^{6}+\frac{22557}{1528}a^{5}-\frac{104923}{6112}a^{4}+\frac{14913}{764}a^{3}-\frac{7257}{382}a^{2}+\frac{4955}{382}a-\frac{948}{191}$ 975/1564672a^23 - 881/391168a^22 + 1393/391168a^21 - 879/391168a^20 - 257/195584a^19 - 655/195584a^18 + 2271/195584a^17 - 1577/195584a^16 - 247/12224a^15 + 6243/195584a^14 - 2003/48896a^13 + 6483/97792a^12 - 56129/391168a^11 + 4537/24448a^10 - 21237/97792a^9 + 22879/48896a^8 - 25165/24448a^7 + 15573/12224a^6 - 13201/12224a^5 + 2749/3056a^4 - 6945/3056a^3 + 1269/382a^2 - 2133/764a - 94/191, 89/48896a^23 - 4093/391168a^22 + 9617/391168a^21 - 3139/97792a^20 + 1585/48896a^19 - 5953/97792a^18 + 3599/24448a^17 - 5609/24448a^16 + 1399/6112a^15 - 11495/48896a^14 + 18883/48896a^13 - 27867/48896a^12 + 15025/24448a^11 - 73337/97792a^10 + 123493/97792a^9 - 15097/12224a^8 - 10459/24448a^7 + 12725/6112a^6 - 154/191a^5 - 192/191a^4 - 8565/3056a^3 + 4489/382a^2 - 10325/764a + 2299/382, 105/1564672a^23 + 293/391168a^22 - 213/97792a^21 + 17/391168a^20 + 1277/195584a^19 - 1863/195584a^18 + 1561/195584a^17 - 2979/195584a^16 + 2835/48896a^15 - 20943/195584a^14 + 6913/48896a^13 - 17009/97792a^12 + 130441/391168a^11 - 11659/24448a^10 + 28269/48896a^9 - 9377/12224a^8 + 36107/24448a^7 - 25345/12224a^6 + 23855/12224a^5 - 4047/3056a^4 + 2285/764a^3 - 6585/1528a^2 + 3079/764a - 223/382, 4835/1564672a^23 - 3327/195584a^22 + 3671/97792a^21 - 17153/391168a^20 + 8015/195584a^19 - 18009/195584a^18 + 44631/195584a^17 - 65469/195584a^16 + 3843/12224a^15 - 64165/195584a^14 + 13899/24448a^13 - 79107/97792a^12 + 322483/391168a^11 - 108097/97792a^10 + 94109/48896a^9 - 38739/24448a^8 - 14923/12224a^7 + 40225/12224a^6 - 7515/12224a^5 - 5887/3056a^4 - 4015/764a^3 + 27643/1528a^2 - 3653/191a + 3065/382, 1459/1564672a^23 - 3643/782336a^22 + 4811/391168a^21 - 10347/391168a^20 + 4631/97792a^19 - 16697/195584a^18 + 29305/195584a^17 - 50551/195584a^16 + 40395/97792a^15 - 125089/195584a^14 + 89705/97792a^13 - 128973/97792a^12 + 723947/391168a^11 - 517497/195584a^10 + 352863/97792a^9 - 235543/48896a^8 + 143021/24448a^7 - 86321/12224a^6 + 96359/12224a^5 - 56385/6112a^4 + 30311/3056a^3 - 7591/764a^2 + 2691/382a - 688/191, 2095/782336a^23 - 4361/391168a^22 + 4529/195584a^21 - 891/24448a^20 + 3085/48896a^19 - 12123/97792a^18 + 21723/97792a^17 - 32971/97792a^16 + 24315/48896a^15 - 72753/97792a^14 + 53001/48896a^13 - 72061/48896a^12 + 399087/195584a^11 - 289023/97792a^10 + 197925/48896a^9 - 232919/48896a^8 + 64181/12224a^7 - 76423/12224a^6 + 11815/1528a^5 - 24875/3056a^4 + 1299/191a^3 - 6369/1528a^2 + 425/191a - 231/382, 2133/1564672a^23 - 1635/782336a^22 + 137/97792a^21 + 7/391168a^20 + 27/3056a^19 - 4677/195584a^18 + 5047/195584a^17 - 3249/195584a^16 + 3829/97792a^15 - 22443/195584a^14 + 16625/97792a^13 - 18621/97792a^12 + 109101/391168a^11 - 97133/195584a^10 + 15151/24448a^9 - 25545/48896a^8 + 21047/24448a^7 - 21649/12224a^6 + 28653/12224a^5 - 8737/6112a^4 + 1129/1528a^3 - 2395/764a^2 + 1095/191a - 1867/382, 2305/782336a^23 - 3189/391168a^22 + 3549/391168a^21 + 29/6112a^20 - 1399/97792a^19 + 1/24448a^18 + 1543/97792a^17 + 5001/97792a^16 - 4615/24448a^15 + 27847/97792a^14 - 17139/48896a^13 + 893/1528a^12 - 186543/195584a^11 + 118061/97792a^10 - 159379/97792a^9 + 156433/48896a^8 - 128883/24448a^7 + 17481/3056a^6 - 13635/3056a^5 + 9169/1528a^4 - 35673/3056a^3 + 22299/1528a^2 - 7185/764a + 405/382, 2365/1564672a^23 - 1307/195584a^22 + 6431/391168a^21 - 10495/391168a^20 + 7571/195584a^19 - 13421/195584a^18 + 26501/195584a^17 - 43087/195584a^16 + 14959/48896a^15 - 78911/195584a^14 + 14437/24448a^13 - 81855/97792a^12 + 445933/391168a^11 - 150583/97792a^10 + 212337/97792a^9 - 16083/6112a^8 + 63001/24448a^7 - 13881/6112a^6 + 33975/12224a^5 - 5999/1528a^4 + 9747/3056a^3 + 879/1528a^2 - 2829/764a + 1453/382, 1825/1564672a^23 - 4831/782336a^22 + 8357/391168a^21 - 18659/391168a^20 + 8401/97792a^19 - 27251/195584a^18 + 48651/195584a^17 - 86649/195584a^16 + 70855/97792a^15 - 207171/195584a^14 + 148529/97792a^13 - 214303/97792a^12 + 1221673/391168a^11 - 843577/195584a^10 + 578501/97792a^9 - 196619/24448a^8 + 30923/3056a^7 - 68547/6112a^6 + 148787/12224a^5 - 88211/6112a^4 + 51911/3056a^3 - 23483/1528a^2 + 6821/764a - 729/382, 139/195584a^23 - 4339/782336a^22 + 3887/195584a^21 - 4559/97792a^20 + 16691/195584a^19 - 14389/97792a^18 + 25545/97792a^17 - 45827/97792a^16 + 75137/97792a^15 - 1757/1528a^14 + 162437/97792a^13 - 58809/24448a^12 + 83721/24448a^11 - 938355/195584a^10 + 161353/24448a^9 - 109841/12224a^8 + 278379/24448a^7 - 80937/6112a^6 + 22557/1528a^5 - 104923/6112a^4 + 14913/764a^3 - 7257/382a^2 + 4955/382*a - 948/191 (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:	$ 25169728.518406376 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 25169728.518406376 \cdot 2}{12\cdot\sqrt{2243572222946525052726785077149696}}\cr\approx \mathstrut & 0.335286196293265 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096)

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 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096, 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 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096);

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 - 4*x^23 + 8*x^22 - 12*x^21 + 24*x^20 - 56*x^19 + 104*x^18 - 152*x^17 + 224*x^16 - 376*x^15 + 608*x^14 - 848*x^13 + 1156*x^12 - 1696*x^11 + 2432*x^10 - 3008*x^9 + 3584*x^8 - 4864*x^7 + 6656*x^6 - 7168*x^5 + 6144*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096);

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

$C_2^3\times S_4$ (as 24T400):

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 192

The 40 conjugacy class representatives for $C_2^3\times S_4$

Character table for $C_2^3\times S_4$

Intermediate fields

$\Q(\sqrt{-3}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-1}) $, 3.3.148.1, $\Q(\zeta_{12})$, 6.6.9462528.1, 6.0.503792.1, 6.0.591408.1, 6.6.8060672.1, 6.0.217638144.1, 6.0.350464.1, 6.6.13602384.1, 12.0.47366361723764736.1, 12.0.47366361723764736.3, 12.0.47366361723764736.2, 12.0.89539436150784.1, 12.0.64974433091584.2, 12.12.47366361723764736.1, 12.0.185024850483456.1

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

Degree 24 siblings:	data not computed
Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

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	${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$	${\href{/padicField/7.6.0.1}{6} }^{4}$	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$	${\href{/padicField/17.2.0.1}{2} }^{12}$	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$	R	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$	R	${\href{/padicField/41.6.0.1}{6} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
$2$ 2	2.12.16.13	$x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$	$6$	$2$	$16$	$D_6$	$[2]_{3}^{2}$
$2$ 2	2.12.16.13	$x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$	$6$	$2$	$16$	$D_6$	$[2]_{3}^{2}$
$3$ 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	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$23$ 23	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$37$ 37	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

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