LMFDB - Number field 24.0.7404154726819150028835278894923776.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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 + 12*y^20 - 4*y^19 - 8*y^18 + 24*y^17 - 56*y^16 + 88*y^15 - 88*y^14 + 56*y^13 - 28*y^12 + 112*y^11 - 352*y^10 + 704*y^9 - 896*y^8 + 768*y^7 - 512*y^6 - 512*y^5 + 3072*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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*x^4 - 6144*x^3 + 8192*x^2 - 8192*x + 4096)

$ x^{24} - 4 x^{23} + 8 x^{22} - 12 x^{21} + 12 x^{20} - 4 x^{19} - 8 x^{18} + 24 x^{17} - 56 x^{16} + \cdots + 4096 $ x^24 - 4*x^23 + 8*x^22 - 12*x^21 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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:	$7404154726819150028835278894923776$ 7404154726819150028835278894923776 $\medspace = 2^{32}\cdot 3^{12}\cdot 31^{4}\cdot 37^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$25.78$	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:	$2^{4/3}3^{1/2}31^{1/2}37^{1/2}\approx 147.81403207845423$
Ramified primes:	$2$, $3$, $31$, $37$ 2, 3, 31, 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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{15}+\frac{1}{4}a^{3}$, $\frac{1}{32}a^{16}+\frac{1}{8}a^{4}$, $\frac{1}{64}a^{17}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{5}{16}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{256}a^{18}-\frac{1}{64}a^{15}-\frac{1}{64}a^{14}+\frac{3}{64}a^{13}+\frac{1}{32}a^{12}-\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{5}{32}a^{7}-\frac{15}{64}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}$, $\frac{1}{512}a^{19}-\frac{1}{128}a^{16}-\frac{1}{128}a^{15}+\frac{3}{128}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{12}+\frac{1}{64}a^{11}-\frac{1}{64}a^{10}+\frac{1}{64}a^{9}-\frac{5}{64}a^{8}-\frac{15}{128}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{512}a^{20}-\frac{1}{128}a^{17}-\frac{1}{128}a^{16}+\frac{3}{128}a^{15}+\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{12}-\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{5}{64}a^{9}-\frac{15}{128}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}-\frac{1}{16}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{1024}a^{21}-\frac{1}{256}a^{17}+\frac{3}{256}a^{16}-\frac{1}{128}a^{15}+\frac{5}{128}a^{14}-\frac{1}{128}a^{13}+\frac{3}{128}a^{12}-\frac{3}{128}a^{11}+\frac{15}{128}a^{10}-\frac{23}{256}a^{9}+\frac{1}{32}a^{8}-\frac{3}{32}a^{7}+\frac{15}{64}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{174080}a^{22}-\frac{33}{87040}a^{21}-\frac{19}{21760}a^{20}+\frac{3}{21760}a^{19}+\frac{47}{43520}a^{18}+\frac{133}{43520}a^{17}+\frac{41}{5440}a^{16}+\frac{327}{21760}a^{15}-\frac{719}{21760}a^{14}+\frac{49}{4352}a^{13}+\frac{7}{21760}a^{12}+\frac{613}{21760}a^{11}+\frac{3853}{43520}a^{10}+\frac{111}{4352}a^{9}-\frac{39}{2720}a^{8}-\frac{2399}{10880}a^{7}+\frac{377}{5440}a^{6}-\frac{329}{1360}a^{5}-\frac{123}{340}a^{4}+\frac{3}{85}a^{3}-\frac{38}{85}a^{2}+\frac{123}{340}a-\frac{83}{170}$, $\frac{1}{348160}a^{23}-\frac{11}{43520}a^{21}-\frac{37}{87040}a^{20}-\frac{67}{87040}a^{19}-\frac{33}{17408}a^{18}+\frac{133}{43520}a^{17}+\frac{271}{43520}a^{16}+\frac{973}{43520}a^{15}-\frac{37}{2560}a^{14}-\frac{1503}{43520}a^{13}-\frac{193}{8704}a^{12}-\frac{3631}{87040}a^{11}+\frac{7}{160}a^{10}-\frac{823}{10880}a^{9}+\frac{99}{2176}a^{8}+\frac{443}{2176}a^{7}+\frac{81}{680}a^{6}+\frac{149}{2720}a^{5}+\frac{363}{1360}a^{4}-\frac{1}{17}a^{3}-\frac{38}{85}a^{2}+\frac{151}{340}a+\frac{33}{85}$ 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, 1/4*a^9 - 1/2*a^3, 1/4*a^10 - 1/2*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, 1/8*a^13 - 1/4*a^7 - 1/2*a^4, 1/8*a^14 - 1/2*a^2, 1/16*a^15 + 1/4*a^3, 1/32*a^16 + 1/8*a^4, 1/64*a^17 - 1/16*a^14 - 1/16*a^13 - 1/16*a^12 - 1/8*a^10 - 1/8*a^9 - 1/8*a^8 + 1/8*a^7 - 1/8*a^6 + 5/16*a^5 - 1/2*a^3, 1/256*a^18 - 1/64*a^15 - 1/64*a^14 + 3/64*a^13 + 1/32*a^12 - 1/32*a^11 + 1/32*a^10 - 1/32*a^9 + 1/32*a^8 - 5/32*a^7 - 15/64*a^6 - 1/2*a^5 - 1/4*a^4 - 1/8*a^3 - 1/2, 1/512*a^19 - 1/128*a^16 - 1/128*a^15 + 3/128*a^14 + 1/64*a^13 - 1/64*a^12 + 1/64*a^11 - 1/64*a^10 + 1/64*a^9 - 5/64*a^8 - 15/128*a^7 - 1/4*a^6 + 3/8*a^5 - 1/16*a^4 - 1/2*a^3 - 1/2*a^2 + 1/4*a, 1/512*a^20 - 1/128*a^17 - 1/128*a^16 + 3/128*a^15 + 1/64*a^14 - 1/64*a^13 + 1/64*a^12 - 1/64*a^11 + 1/64*a^10 - 5/64*a^9 - 15/128*a^8 - 1/4*a^7 - 1/8*a^6 - 1/16*a^5 - 1/2*a^4 - 1/2*a^3 + 1/4*a^2, 1/1024*a^21 - 1/256*a^17 + 3/256*a^16 - 1/128*a^15 + 5/128*a^14 - 1/128*a^13 + 3/128*a^12 - 3/128*a^11 + 15/128*a^10 - 23/256*a^9 + 1/32*a^8 - 3/32*a^7 + 15/64*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2, 1/174080*a^22 - 33/87040*a^21 - 19/21760*a^20 + 3/21760*a^19 + 47/43520*a^18 + 133/43520*a^17 + 41/5440*a^16 + 327/21760*a^15 - 719/21760*a^14 + 49/4352*a^13 + 7/21760*a^12 + 613/21760*a^11 + 3853/43520*a^10 + 111/4352*a^9 - 39/2720*a^8 - 2399/10880*a^7 + 377/5440*a^6 - 329/1360*a^5 - 123/340*a^4 + 3/85*a^3 - 38/85*a^2 + 123/340*a - 83/170, 1/348160*a^23 - 11/43520*a^21 - 37/87040*a^20 - 67/87040*a^19 - 33/17408*a^18 + 133/43520*a^17 + 271/43520*a^16 + 973/43520*a^15 - 37/2560*a^14 - 1503/43520*a^13 - 193/8704*a^12 - 3631/87040*a^11 + 7/160*a^10 - 823/10880*a^9 + 99/2176*a^8 + 443/2176*a^7 + 81/680*a^6 + 149/2720*a^5 + 363/1360*a^4 - 1/17*a^3 - 38/85*a^2 + 151/340*a + 33/85

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

$C_{3}$, which has order $3$ (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{69}{348160} a^{23} - \frac{603}{174080} a^{22} + \frac{1527}{87040} a^{21} - \frac{2619}{87040} a^{20} + \frac{1977}{87040} a^{19} - \frac{417}{87040} a^{18} - \frac{159}{10880} a^{17} + \frac{1797}{43520} a^{16} - \frac{207}{2560} a^{15} + \frac{1427}{8704} a^{14} - \frac{12243}{43520} a^{13} + \frac{5103}{43520} a^{12} + \frac{20463}{87040} a^{11} - \frac{531}{8704} a^{10} - \frac{19641}{21760} a^{9} + \frac{17583}{10880} a^{8} - \frac{15663}{10880} a^{7} + \frac{7947}{5440} a^{6} - \frac{3147}{2720} a^{5} + \frac{27}{680} a^{4} + \frac{321}{85} a^{3} - \frac{2223}{170} a^{2} + \frac{7467}{340} a - \frac{489}{34} $ -(69)/(348160)a^(23) - (603)/(174080)a^(22) + (1527)/(87040)a^(21) - (2619)/(87040)a^(20) + (1977)/(87040)a^(19) - (417)/(87040)a^(18) - (159)/(10880)a^(17) + (1797)/(43520)a^(16) - (207)/(2560)a^(15) + (1427)/(8704)a^(14) - (12243)/(43520)a^(13) + (5103)/(43520)a^(12) + (20463)/(87040)a^(11) - (531)/(8704)a^(10) - (19641)/(21760)a^(9) + (17583)/(10880)a^(8) - (15663)/(10880)a^(7) + (7947)/(5440)a^(6) - (3147)/(2720)a^(5) + (27)/(680)a^(4) + (321)/(85)a^(3) - (2223)/(170)a^(2) + (7467)/(340)*a - (489)/(34) (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:	$\frac{1811}{348160}a^{23}-\frac{57}{10880}a^{22}-\frac{1303}{43520}a^{21}+\frac{1107}{17408}a^{20}-\frac{4441}{87040}a^{19}+\frac{2037}{87040}a^{18}+\frac{227}{43520}a^{17}-\frac{363}{8704}a^{16}+\frac{503}{8704}a^{15}-\frac{12623}{43520}a^{14}+\frac{31267}{43520}a^{13}-\frac{14823}{43520}a^{12}-\frac{13705}{17408}a^{11}+\frac{4271}{5440}a^{10}+\frac{16987}{10880}a^{9}-\frac{119}{40}a^{8}+\frac{27233}{10880}a^{7}-\frac{16471}{5440}a^{6}+\frac{1811}{544}a^{5}-\frac{5533}{1360}a^{4}+\frac{447}{680}a^{3}+\frac{7597}{340}a^{2}-\frac{3307}{68}a+\frac{5077}{170}$, $\frac{257}{10240}a^{23}-\frac{833}{10240}a^{22}+\frac{821}{5120}a^{21}-\frac{53}{256}a^{20}+\frac{323}{2560}a^{19}+\frac{21}{640}a^{18}-\frac{527}{2560}a^{17}+\frac{125}{256}a^{16}-\frac{69}{64}a^{15}+\frac{947}{640}a^{14}-\frac{217}{160}a^{13}+\frac{23}{160}a^{12}+\frac{115}{512}a^{11}+\frac{7387}{2560}a^{10}-\frac{10619}{1280}a^{9}+\frac{8303}{640}a^{8}-\frac{7383}{640}a^{7}+\frac{3251}{320}a^{6}-\frac{79}{16}a^{5}-\frac{363}{20}a^{4}+\frac{1339}{20}a^{3}-\frac{2337}{20}a^{2}+\frac{547}{4}a-\frac{877}{10}$, $\frac{1227}{348160}a^{23}-\frac{3157}{174080}a^{22}+\frac{3747}{87040}a^{21}-\frac{4537}{87040}a^{20}+\frac{2477}{87040}a^{19}+\frac{587}{87040}a^{18}-\frac{203}{4352}a^{17}+\frac{4901}{43520}a^{16}-\frac{10607}{43520}a^{15}+\frac{17443}{43520}a^{14}-\frac{16951}{43520}a^{13}-\frac{1533}{43520}a^{12}+\frac{22039}{87040}a^{11}+\frac{29567}{43520}a^{10}-\frac{49457}{21760}a^{9}+\frac{4149}{1360}a^{8}-\frac{28827}{10880}a^{7}+\frac{14097}{5440}a^{6}-\frac{3931}{2720}a^{5}-\frac{59}{17}a^{4}+\frac{77}{5}a^{3}-\frac{2103}{68}a^{2}+\frac{11931}{340}a-\frac{3107}{170}$, $\frac{2581}{348160}a^{23}-\frac{83}{4352}a^{22}+\frac{503}{87040}a^{21}+\frac{1953}{87040}a^{20}-\frac{2477}{87040}a^{19}+\frac{515}{17408}a^{18}-\frac{1557}{43520}a^{17}+\frac{2001}{43520}a^{16}-\frac{5067}{43520}a^{15}+\frac{1631}{43520}a^{14}+\frac{16577}{43520}a^{13}-\frac{3481}{8704}a^{12}-\frac{46451}{87040}a^{11}+\frac{14041}{10880}a^{10}-\frac{7561}{21760}a^{9}-\frac{691}{1088}a^{8}+\frac{35}{136}a^{7}-\frac{5427}{5440}a^{6}+\frac{6549}{2720}a^{5}-\frac{4321}{680}a^{4}+\frac{25}{2}a^{3}-\frac{1317}{340}a^{2}-\frac{1661}{85}a+\frac{3141}{170}$, $\frac{301}{43520}a^{23}-\frac{567}{43520}a^{22}+\frac{1043}{87040}a^{21}-\frac{609}{43520}a^{20}+\frac{329}{43520}a^{19}+\frac{287}{21760}a^{18}-\frac{127}{4352}a^{17}+\frac{1337}{21760}a^{16}-\frac{1737}{10880}a^{15}+\frac{489}{5440}a^{14}+\frac{139}{10880}a^{13}+\frac{97}{10880}a^{12}-\frac{1579}{5440}a^{11}+\frac{2891}{5440}a^{10}-\frac{13253}{21760}a^{9}+\frac{14153}{10880}a^{8}-\frac{12103}{10880}a^{7}+\frac{203}{340}a^{6}+\frac{107}{340}a^{5}-\frac{1151}{272}a^{4}+\frac{7093}{680}a^{3}-\frac{567}{68}a^{2}+\frac{2449}{340}a-\frac{834}{85}$, $\frac{447}{43520}a^{23}-\frac{5147}{174080}a^{22}+\frac{2707}{43520}a^{21}-\frac{385}{4352}a^{20}+\frac{623}{10880}a^{19}+\frac{441}{43520}a^{18}-\frac{3413}{43520}a^{17}+\frac{865}{4352}a^{16}-\frac{1827}{4352}a^{15}+\frac{12271}{21760}a^{14}-\frac{13389}{21760}a^{13}+\frac{3321}{21760}a^{12}+\frac{433}{4352}a^{11}+\frac{39093}{43520}a^{10}-\frac{35363}{10880}a^{9}+\frac{437}{80}a^{8}-\frac{52457}{10880}a^{7}+\frac{22719}{5440}a^{6}-\frac{575}{272}a^{5}-\frac{4689}{680}a^{4}+\frac{17077}{680}a^{3}-\frac{7679}{170}a^{2}+\frac{4057}{68}a-\frac{6583}{170}$, $\frac{3369}{348160}a^{23}-\frac{2187}{43520}a^{22}+\frac{2429}{21760}a^{21}-\frac{2087}{17408}a^{20}+\frac{5391}{87040}a^{19}+\frac{1863}{87040}a^{18}-\frac{5487}{43520}a^{17}+\frac{2519}{8704}a^{16}-\frac{303}{512}a^{15}+\frac{44623}{43520}a^{14}-\frac{38207}{43520}a^{13}-\frac{12317}{43520}a^{12}+\frac{9285}{17408}a^{11}+\frac{22523}{10880}a^{10}-\frac{31811}{5440}a^{9}+\frac{38659}{5440}a^{8}-\frac{18177}{2720}a^{7}+\frac{34061}{5440}a^{6}-\frac{1505}{544}a^{5}-\frac{6371}{680}a^{4}+\frac{27333}{680}a^{3}-\frac{27037}{340}a^{2}+\frac{1390}{17}a-\frac{5517}{170}$, $\frac{5331}{348160}a^{23}-\frac{959}{21760}a^{22}+\frac{3209}{87040}a^{21}+\frac{53}{17408}a^{20}-\frac{2651}{87040}a^{19}+\frac{4697}{87040}a^{18}-\frac{3823}{43520}a^{17}+\frac{1319}{8704}a^{16}-\frac{189}{512}a^{15}+\frac{14177}{43520}a^{14}+\frac{17727}{43520}a^{13}-\frac{29763}{43520}a^{12}-\frac{13233}{17408}a^{11}+\frac{28797}{10880}a^{10}-\frac{43851}{21760}a^{9}+\frac{9487}{10880}a^{8}-\frac{757}{680}a^{7}+\frac{39}{5440}a^{6}+\frac{1551}{544}a^{5}-\frac{9059}{680}a^{4}+\frac{10671}{340}a^{3}-\frac{4409}{170}a^{2}-\frac{172}{17}a+\frac{3047}{170}$, $\frac{29}{2720}a^{23}-\frac{4143}{174080}a^{22}+\frac{7}{136}a^{21}-\frac{1859}{21760}a^{20}+\frac{65}{1088}a^{19}+\frac{369}{43520}a^{18}-\frac{3093}{43520}a^{17}+\frac{3979}{21760}a^{16}-\frac{8723}{21760}a^{15}+\frac{10243}{21760}a^{14}-\frac{13433}{21760}a^{13}+\frac{7629}{21760}a^{12}+\frac{143}{21760}a^{11}+\frac{21849}{43520}a^{10}-\frac{14283}{5440}a^{9}+\frac{7401}{1360}a^{8}-\frac{48413}{10880}a^{7}+\frac{4133}{1088}a^{6}-\frac{1347}{680}a^{5}-\frac{2187}{340}a^{4}+\frac{15163}{680}a^{3}-\frac{3257}{85}a^{2}+\frac{20153}{340}a-\frac{8009}{170}$, $\frac{437}{348160}a^{23}-\frac{1331}{174080}a^{22}+\frac{311}{21760}a^{21}-\frac{1203}{87040}a^{20}+\frac{819}{87040}a^{19}+\frac{321}{87040}a^{18}-\frac{47}{2720}a^{17}+\frac{1629}{43520}a^{16}-\frac{3613}{43520}a^{15}+\frac{1149}{8704}a^{14}-\frac{3741}{43520}a^{13}-\frac{2079}{43520}a^{12}+\frac{1441}{87040}a^{11}+\frac{2777}{8704}a^{10}-\frac{2019}{2720}a^{9}+\frac{4533}{5440}a^{8}-\frac{10861}{10880}a^{7}+\frac{5279}{5440}a^{6}-\frac{889}{2720}a^{5}-\frac{219}{170}a^{4}+\frac{4041}{680}a^{3}-\frac{3427}{340}a^{2}+\frac{2829}{340}a-\frac{7}{2}$, $\frac{13}{640}a^{23}-\frac{9723}{174080}a^{22}+\frac{95}{1088}a^{21}-\frac{4543}{43520}a^{20}+\frac{535}{8704}a^{19}+\frac{1339}{43520}a^{18}-\frac{5613}{43520}a^{17}+\frac{6539}{21760}a^{16}-\frac{15143}{21760}a^{15}+\frac{17233}{21760}a^{14}-\frac{10873}{21760}a^{13}-\frac{231}{21760}a^{12}-\frac{7717}{21760}a^{11}+\frac{97149}{43520}a^{10}-\frac{24473}{5440}a^{9}+\frac{77653}{10880}a^{8}-\frac{35089}{5440}a^{7}+\frac{353}{68}a^{6}-\frac{1057}{680}a^{5}-\frac{19413}{1360}a^{4}+\frac{3811}{85}a^{3}-\frac{21453}{340}a^{2}+\frac{5492}{85}a-\frac{3977}{85}$ 1811/348160a^23 - 57/10880a^22 - 1303/43520a^21 + 1107/17408a^20 - 4441/87040a^19 + 2037/87040a^18 + 227/43520a^17 - 363/8704a^16 + 503/8704a^15 - 12623/43520a^14 + 31267/43520a^13 - 14823/43520a^12 - 13705/17408a^11 + 4271/5440a^10 + 16987/10880a^9 - 119/40a^8 + 27233/10880a^7 - 16471/5440a^6 + 1811/544a^5 - 5533/1360a^4 + 447/680a^3 + 7597/340a^2 - 3307/68a + 5077/170, 257/10240a^23 - 833/10240a^22 + 821/5120a^21 - 53/256a^20 + 323/2560a^19 + 21/640a^18 - 527/2560a^17 + 125/256a^16 - 69/64a^15 + 947/640a^14 - 217/160a^13 + 23/160a^12 + 115/512a^11 + 7387/2560a^10 - 10619/1280a^9 + 8303/640a^8 - 7383/640a^7 + 3251/320a^6 - 79/16a^5 - 363/20a^4 + 1339/20a^3 - 2337/20a^2 + 547/4a - 877/10, 1227/348160a^23 - 3157/174080a^22 + 3747/87040a^21 - 4537/87040a^20 + 2477/87040a^19 + 587/87040a^18 - 203/4352a^17 + 4901/43520a^16 - 10607/43520a^15 + 17443/43520a^14 - 16951/43520a^13 - 1533/43520a^12 + 22039/87040a^11 + 29567/43520a^10 - 49457/21760a^9 + 4149/1360a^8 - 28827/10880a^7 + 14097/5440a^6 - 3931/2720a^5 - 59/17a^4 + 77/5a^3 - 2103/68a^2 + 11931/340a - 3107/170, 2581/348160a^23 - 83/4352a^22 + 503/87040a^21 + 1953/87040a^20 - 2477/87040a^19 + 515/17408a^18 - 1557/43520a^17 + 2001/43520a^16 - 5067/43520a^15 + 1631/43520a^14 + 16577/43520a^13 - 3481/8704a^12 - 46451/87040a^11 + 14041/10880a^10 - 7561/21760a^9 - 691/1088a^8 + 35/136a^7 - 5427/5440a^6 + 6549/2720a^5 - 4321/680a^4 + 25/2a^3 - 1317/340a^2 - 1661/85a + 3141/170, 301/43520a^23 - 567/43520a^22 + 1043/87040a^21 - 609/43520a^20 + 329/43520a^19 + 287/21760a^18 - 127/4352a^17 + 1337/21760a^16 - 1737/10880a^15 + 489/5440a^14 + 139/10880a^13 + 97/10880a^12 - 1579/5440a^11 + 2891/5440a^10 - 13253/21760a^9 + 14153/10880a^8 - 12103/10880a^7 + 203/340a^6 + 107/340a^5 - 1151/272a^4 + 7093/680a^3 - 567/68a^2 + 2449/340a - 834/85, 447/43520a^23 - 5147/174080a^22 + 2707/43520a^21 - 385/4352a^20 + 623/10880a^19 + 441/43520a^18 - 3413/43520a^17 + 865/4352a^16 - 1827/4352a^15 + 12271/21760a^14 - 13389/21760a^13 + 3321/21760a^12 + 433/4352a^11 + 39093/43520a^10 - 35363/10880a^9 + 437/80a^8 - 52457/10880a^7 + 22719/5440a^6 - 575/272a^5 - 4689/680a^4 + 17077/680a^3 - 7679/170a^2 + 4057/68a - 6583/170, 3369/348160a^23 - 2187/43520a^22 + 2429/21760a^21 - 2087/17408a^20 + 5391/87040a^19 + 1863/87040a^18 - 5487/43520a^17 + 2519/8704a^16 - 303/512a^15 + 44623/43520a^14 - 38207/43520a^13 - 12317/43520a^12 + 9285/17408a^11 + 22523/10880a^10 - 31811/5440a^9 + 38659/5440a^8 - 18177/2720a^7 + 34061/5440a^6 - 1505/544a^5 - 6371/680a^4 + 27333/680a^3 - 27037/340a^2 + 1390/17a - 5517/170, 5331/348160a^23 - 959/21760a^22 + 3209/87040a^21 + 53/17408a^20 - 2651/87040a^19 + 4697/87040a^18 - 3823/43520a^17 + 1319/8704a^16 - 189/512a^15 + 14177/43520a^14 + 17727/43520a^13 - 29763/43520a^12 - 13233/17408a^11 + 28797/10880a^10 - 43851/21760a^9 + 9487/10880a^8 - 757/680a^7 + 39/5440a^6 + 1551/544a^5 - 9059/680a^4 + 10671/340a^3 - 4409/170a^2 - 172/17a + 3047/170, 29/2720a^23 - 4143/174080a^22 + 7/136a^21 - 1859/21760a^20 + 65/1088a^19 + 369/43520a^18 - 3093/43520a^17 + 3979/21760a^16 - 8723/21760a^15 + 10243/21760a^14 - 13433/21760a^13 + 7629/21760a^12 + 143/21760a^11 + 21849/43520a^10 - 14283/5440a^9 + 7401/1360a^8 - 48413/10880a^7 + 4133/1088a^6 - 1347/680a^5 - 2187/340a^4 + 15163/680a^3 - 3257/85a^2 + 20153/340a - 8009/170, 437/348160a^23 - 1331/174080a^22 + 311/21760a^21 - 1203/87040a^20 + 819/87040a^19 + 321/87040a^18 - 47/2720a^17 + 1629/43520a^16 - 3613/43520a^15 + 1149/8704a^14 - 3741/43520a^13 - 2079/43520a^12 + 1441/87040a^11 + 2777/8704a^10 - 2019/2720a^9 + 4533/5440a^8 - 10861/10880a^7 + 5279/5440a^6 - 889/2720a^5 - 219/170a^4 + 4041/680a^3 - 3427/340a^2 + 2829/340a - 7/2, 13/640a^23 - 9723/174080a^22 + 95/1088a^21 - 4543/43520a^20 + 535/8704a^19 + 1339/43520a^18 - 5613/43520a^17 + 6539/21760a^16 - 15143/21760a^15 + 17233/21760a^14 - 10873/21760a^13 - 231/21760a^12 - 7717/21760a^11 + 97149/43520a^10 - 24473/5440a^9 + 77653/10880a^8 - 35089/5440a^7 + 353/68a^6 - 1057/680a^5 - 19413/1360a^4 + 3811/85a^3 - 21453/340a^2 + 5492/85*a - 3977/85 (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:	$ 39316667.48906936 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 39316667.48906936 \cdot 3}{12\cdot\sqrt{7404154726819150028835278894923776}}\cr\approx \mathstrut & 0.432451477932467 \end{aligned}\] (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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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 + 12*x^20 - 4*x^19 - 8*x^18 + 24*x^17 - 56*x^16 + 88*x^15 - 88*x^14 + 56*x^13 - 28*x^12 + 112*x^11 - 352*x^10 + 704*x^9 - 896*x^8 + 768*x^7 - 512*x^6 - 512*x^5 + 3072*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, $\Q(\zeta_{12})$, 6.0.293338368.1, 6.6.9462528.1, 6.6.18333648.1, 6.0.591408.1, 6.0.679024.1, 6.0.350464.1, 6.6.10864384.1, 12.0.118034839699456.1, 12.0.86047398140903424.2, 12.0.336122648987904.1, 12.0.86047398140903424.3, 12.0.89539436150784.1, 12.12.86047398140903424.1, 12.0.86047398140903424.1

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	${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$	${\href{/padicField/7.6.0.1}{6} }^{4}$	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$	${\href{/padicField/19.2.0.1}{2} }^{12}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	R	R	${\href{/padicField/41.6.0.1}{6} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$31$ 31	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$37$ 37	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.2.0.1	$x^{2} + 33 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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