LMFDB - Number field 24.0.1041229780068396944496497143054336.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 4096)

gp: K = bnfinit(y^24 + 9*y^22 + 42*y^20 + 139*y^18 + 376*y^16 + 896*y^14 + 1905*y^12 + 3584*y^10 + 6016*y^8 + 8896*y^6 + 10752*y^4 + 9216*y^2 + 4096, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 4096);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 4096)

$ x^{24} + 9 x^{22} + 42 x^{20} + 139 x^{18} + 376 x^{16} + 896 x^{14} + 1905 x^{12} + 3584 x^{10} + \cdots + 4096 $ x^24 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 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:	$1041229780068396944496497143054336$ 1041229780068396944496497143054336 $\medspace = 2^{24}\cdot 7^{20}\cdot 167^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$23.75$	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:	$2\cdot 7^{5/6}167^{1/2}\approx 130.80869046079246$
Ramified primes:	$2$, $7$, $167$ 2, 7, 167	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{13}-\frac{1}{4}a^{9}-\frac{3}{8}a^{7}+\frac{3}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{10}-\frac{3}{16}a^{8}-\frac{5}{16}a^{6}+\frac{1}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{8}a^{11}-\frac{3}{16}a^{9}-\frac{5}{16}a^{7}+\frac{1}{16}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{12}+\frac{1}{16}a^{10}+\frac{7}{16}a^{8}-\frac{3}{16}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{15}-\frac{1}{16}a^{13}-\frac{1}{32}a^{11}-\frac{1}{16}a^{9}+\frac{3}{16}a^{7}-\frac{13}{32}a^{5}+\frac{3}{8}a^{3}$, $\frac{1}{64}a^{18}+\frac{1}{64}a^{16}-\frac{1}{32}a^{14}-\frac{5}{64}a^{12}-\frac{1}{8}a^{10}+\frac{7}{16}a^{8}+\frac{21}{64}a^{6}-\frac{3}{16}a^{4}$, $\frac{1}{128}a^{19}+\frac{1}{128}a^{17}+\frac{1}{64}a^{15}-\frac{5}{128}a^{13}+\frac{49}{128}a^{7}-\frac{5}{16}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{1024}a^{20}+\frac{1}{1024}a^{18}+\frac{9}{512}a^{16}-\frac{21}{1024}a^{14}-\frac{3}{64}a^{10}-\frac{15}{1024}a^{8}+\frac{47}{128}a^{6}-\frac{5}{64}a^{4}-\frac{1}{16}a^{2}+\frac{1}{4}$, $\frac{1}{2048}a^{21}+\frac{1}{2048}a^{19}+\frac{9}{1024}a^{17}-\frac{21}{2048}a^{15}+\frac{13}{128}a^{11}-\frac{271}{2048}a^{9}+\frac{15}{256}a^{7}+\frac{27}{128}a^{5}+\frac{3}{32}a^{3}-\frac{3}{8}a$, $\frac{1}{1851392}a^{22}-\frac{667}{1851392}a^{20}-\frac{5957}{925696}a^{18}-\frac{45837}{1851392}a^{16}-\frac{6157}{462848}a^{14}-\frac{819}{115712}a^{12}-\frac{165327}{1851392}a^{10}+\frac{228135}{462848}a^{8}-\frac{7879}{115712}a^{6}+\frac{4117}{14464}a^{4}+\frac{77}{226}a^{2}-\frac{567}{1808}$, $\frac{1}{3702784}a^{23}-\frac{667}{3702784}a^{21}-\frac{5957}{1851392}a^{19}-\frac{45837}{3702784}a^{17}-\frac{6157}{925696}a^{15}-\frac{819}{231424}a^{13}+\frac{297521}{3702784}a^{11}+\frac{112423}{925696}a^{9}+\frac{78905}{231424}a^{7}+\frac{11349}{28928}a^{5}+\frac{267}{904}a^{3}-\frac{567}{3616}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/2*a^9 - 1/2*a^7 - 1/2*a^5 - 1/2*a, 1/4*a^10 - 1/4*a^8 - 1/4*a^6 - 1/2*a^4 + 1/4*a^2, 1/4*a^11 - 1/4*a^9 - 1/4*a^7 - 1/2*a^5 + 1/4*a^3, 1/4*a^12 - 1/2*a^8 + 1/4*a^6 - 1/4*a^4 + 1/4*a^2, 1/8*a^13 - 1/4*a^9 - 3/8*a^7 + 3/8*a^5 + 1/8*a^3 - 1/2*a, 1/16*a^14 - 1/8*a^10 - 3/16*a^8 - 5/16*a^6 + 1/16*a^4 + 1/4*a^2, 1/16*a^15 - 1/8*a^11 - 3/16*a^9 - 5/16*a^7 + 1/16*a^5 + 1/4*a^3, 1/16*a^16 - 1/8*a^12 + 1/16*a^10 + 7/16*a^8 - 3/16*a^6 - 1/4*a^4 + 1/4*a^2, 1/32*a^17 - 1/32*a^15 - 1/16*a^13 - 1/32*a^11 - 1/16*a^9 + 3/16*a^7 - 13/32*a^5 + 3/8*a^3, 1/64*a^18 + 1/64*a^16 - 1/32*a^14 - 5/64*a^12 - 1/8*a^10 + 7/16*a^8 + 21/64*a^6 - 3/16*a^4, 1/128*a^19 + 1/128*a^17 + 1/64*a^15 - 5/128*a^13 + 49/128*a^7 - 5/16*a^5 - 1/4*a^3, 1/1024*a^20 + 1/1024*a^18 + 9/512*a^16 - 21/1024*a^14 - 3/64*a^10 - 15/1024*a^8 + 47/128*a^6 - 5/64*a^4 - 1/16*a^2 + 1/4, 1/2048*a^21 + 1/2048*a^19 + 9/1024*a^17 - 21/2048*a^15 + 13/128*a^11 - 271/2048*a^9 + 15/256*a^7 + 27/128*a^5 + 3/32*a^3 - 3/8*a, 1/1851392*a^22 - 667/1851392*a^20 - 5957/925696*a^18 - 45837/1851392*a^16 - 6157/462848*a^14 - 819/115712*a^12 - 165327/1851392*a^10 + 228135/462848*a^8 - 7879/115712*a^6 + 4117/14464*a^4 + 77/226*a^2 - 567/1808, 1/3702784*a^23 - 667/3702784*a^21 - 5957/1851392*a^19 - 45837/3702784*a^17 - 6157/925696*a^15 - 819/231424*a^13 + 297521/3702784*a^11 + 112423/925696*a^9 + 78905/231424*a^7 + 11349/28928*a^5 + 267/904*a^3 - 567/3616*a

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

$C_{4}$, which has order $4$ (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{18113}{3702784} a^{23} + \frac{126245}{3702784} a^{21} + \frac{248187}{1851392} a^{19} + \frac{1463347}{3702784} a^{17} + \frac{921395}{925696} a^{15} + \frac{520797}{231424} a^{13} + \frac{16605425}{3702784} a^{11} + \frac{7293415}{925696} a^{9} + \frac{2866025}{231424} a^{7} + \frac{476669}{28928} a^{5} + \frac{1850}{113} a^{3} + \frac{31913}{3616} a $ (18113)/(3702784)a^(23) + (126245)/(3702784)a^(21) + (248187)/(1851392)a^(19) + (1463347)/(3702784)a^(17) + (921395)/(925696)a^(15) + (520797)/(231424)a^(13) + (16605425)/(3702784)a^(11) + (7293415)/(925696)a^(9) + (2866025)/(231424)a^(7) + (476669)/(28928)a^(5) + (1850)/(113)a^(3) + (31913)/(3616)a (order $28$)	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:	$\frac{6757}{1851392}a^{22}+\frac{61897}{1851392}a^{20}+\frac{145599}{925696}a^{18}+\frac{904639}{1851392}a^{16}+\frac{583667}{462848}a^{14}+\frac{340209}{115712}a^{12}+\frac{11279957}{1851392}a^{10}+\frac{5047647}{462848}a^{8}+\frac{2027637}{115712}a^{6}+\frac{359795}{14464}a^{4}+\frac{48773}{1808}a^{2}+\frac{28409}{1808}$, $\frac{10997}{1851392}a^{22}+\frac{70569}{1851392}a^{20}+\frac{128503}{925696}a^{18}+\frac{724111}{1851392}a^{16}+\frac{445839}{462848}a^{14}+\frac{246801}{115712}a^{12}+\frac{7602917}{1851392}a^{10}+\frac{3259731}{462848}a^{8}+\frac{1263213}{115712}a^{6}+\frac{200305}{14464}a^{4}+\frac{2659}{226}a^{2}+\frac{7725}{1808}$, $\frac{2121}{3702784}a^{23}-\frac{6275}{3702784}a^{21}-\frac{41173}{1851392}a^{19}-\frac{300629}{3702784}a^{17}-\frac{203665}{925696}a^{15}-\frac{127979}{231424}a^{13}-\frac{4592903}{3702784}a^{11}-\frac{2190645}{925696}a^{9}-\frac{901303}{231424}a^{7}-\frac{178345}{28928}a^{5}-\frac{27203}{3616}a^{3}-\frac{17011}{3616}a-1$, $\frac{153}{1851392}a^{23}+\frac{1909}{1851392}a^{21}+\frac{8399}{925696}a^{19}+\frac{65259}{1851392}a^{17}+\frac{45373}{462848}a^{15}+\frac{26565}{115712}a^{13}+\frac{957129}{1851392}a^{11}+\frac{466549}{462848}a^{9}+\frac{196165}{115712}a^{7}+\frac{18043}{7232}a^{5}+\frac{12329}{3616}a^{3}+\frac{5231}{1808}a-1$, $\frac{5449}{3702784}a^{23}+\frac{23101}{3702784}a^{21}+\frac{32779}{1851392}a^{19}+\frac{168491}{3702784}a^{17}+\frac{99647}{925696}a^{15}+\frac{50037}{231424}a^{13}+\frac{1426425}{3702784}a^{11}+\frac{588091}{925696}a^{9}+\frac{207113}{231424}a^{7}+\frac{27215}{28928}a^{5}+\frac{1953}{3616}a^{3}+\frac{1645}{3616}a-1$, $\frac{4061}{1851392}a^{22}+\frac{6929}{1851392}a^{20}-\frac{23841}{925696}a^{18}-\frac{231033}{1851392}a^{16}-\frac{164369}{462848}a^{14}-\frac{109527}{115712}a^{12}-\frac{4255411}{1851392}a^{10}-\frac{2104541}{462848}a^{8}-\frac{888171}{115712}a^{6}-\frac{182499}{14464}a^{4}-\frac{15377}{904}a^{2}-\frac{19987}{1808}$, $\frac{1387}{462848}a^{22}+\frac{11867}{462848}a^{20}+\frac{23931}{231424}a^{18}+\frac{138905}{462848}a^{16}+\frac{21637}{28928}a^{14}+\frac{49191}{28928}a^{12}+\frac{1544123}{462848}a^{10}+\frac{333955}{57856}a^{8}+\frac{264025}{28928}a^{6}+\frac{86545}{7232}a^{4}+\frac{19319}{1808}a^{2}+\frac{380}{113}$, $\frac{7085}{3702784}a^{23}+\frac{4613}{1851392}a^{22}+\frac{79969}{3702784}a^{21}+\frac{31081}{1851392}a^{20}+\frac{175983}{1851392}a^{19}+\frac{51679}{925696}a^{18}+\frac{1039223}{3702784}a^{17}+\frac{245407}{1851392}a^{16}+\frac{661903}{925696}a^{15}+\frac{148339}{462848}a^{14}+\frac{380745}{231424}a^{13}+\frac{76609}{115712}a^{12}+\frac{12334109}{3702784}a^{11}+\frac{2173941}{1851392}a^{10}+\frac{5423843}{925696}a^{9}+\frac{828191}{462848}a^{8}+\frac{2139957}{231424}a^{7}+\frac{277925}{115712}a^{6}+\frac{369765}{28928}a^{5}+\frac{36875}{14464}a^{4}+\frac{22863}{1808}a^{3}-\frac{455}{1808}a^{2}+\frac{20973}{3616}a-\frac{5271}{1808}$, $\frac{12637}{1851392}a^{23}+\frac{14655}{1851392}a^{22}+\frac{101265}{1851392}a^{21}+\frac{96795}{1851392}a^{20}+\frac{207199}{925696}a^{19}+\frac{167589}{925696}a^{18}+\frac{1219847}{1851392}a^{17}+\frac{907085}{1851392}a^{16}+\frac{774191}{462848}a^{15}+\frac{557885}{462848}a^{14}+\frac{440441}{115712}a^{13}+\frac{302803}{115712}a^{12}+\frac{14059725}{1851392}a^{11}+\frac{9167631}{1851392}a^{10}+\frac{6174435}{462848}a^{9}+\frac{3821289}{462848}a^{8}+\frac{2427781}{115712}a^{7}+\frac{1445767}{115712}a^{6}+\frac{410841}{14464}a^{5}+\frac{218635}{14464}a^{4}+\frac{6221}{226}a^{3}+\frac{4667}{452}a^{2}+\frac{24333}{1808}a+\frac{3799}{1808}$, $\frac{1745}{1851392}a^{23}-\frac{15491}{1851392}a^{22}+\frac{14901}{1851392}a^{21}-\frac{88815}{1851392}a^{20}+\frac{22731}{925696}a^{19}-\frac{148689}{925696}a^{18}+\frac{116067}{1851392}a^{17}-\frac{830169}{1851392}a^{16}+\frac{69683}{462848}a^{15}-\frac{509193}{462848}a^{14}+\frac{36229}{115712}a^{13}-\frac{276231}{115712}a^{12}+\frac{1015809}{1851392}a^{11}-\frac{8420115}{1851392}a^{10}+\frac{398855}{462848}a^{9}-\frac{3581189}{462848}a^{8}+\frac{132065}{115712}a^{7}-\frac{1374955}{115712}a^{6}+\frac{17253}{14464}a^{5}-\frac{208975}{14464}a^{4}-\frac{307}{904}a^{3}-\frac{5269}{452}a^{2}-\frac{2247}{1808}a-\frac{12523}{1808}$, $\frac{11901}{1851392}a^{23}+\frac{8877}{925696}a^{22}+\frac{78705}{1851392}a^{21}+\frac{66233}{925696}a^{20}+\frac{147487}{925696}a^{19}+\frac{137051}{462848}a^{18}+\frac{849767}{1851392}a^{17}+\frac{814375}{925696}a^{16}+\frac{530815}{462848}a^{15}+\frac{514753}{231424}a^{14}+\frac{297425}{115712}a^{13}+\frac{293505}{57856}a^{12}+\frac{9325037}{1851392}a^{11}+\frac{9441437}{925696}a^{10}+\frac{4069715}{462848}a^{9}+\frac{4153689}{231424}a^{8}+\frac{1603117}{115712}a^{7}+\frac{1645425}{57856}a^{6}+\frac{263133}{14464}a^{5}+\frac{140025}{3616}a^{4}+\frac{7691}{452}a^{3}+\frac{70045}{1808}a^{2}+\frac{17669}{1808}a+\frac{18067}{904}$ 6757/1851392a^22 + 61897/1851392a^20 + 145599/925696a^18 + 904639/1851392a^16 + 583667/462848a^14 + 340209/115712a^12 + 11279957/1851392a^10 + 5047647/462848a^8 + 2027637/115712a^6 + 359795/14464a^4 + 48773/1808a^2 + 28409/1808, 10997/1851392a^22 + 70569/1851392a^20 + 128503/925696a^18 + 724111/1851392a^16 + 445839/462848a^14 + 246801/115712a^12 + 7602917/1851392a^10 + 3259731/462848a^8 + 1263213/115712a^6 + 200305/14464a^4 + 2659/226a^2 + 7725/1808, 2121/3702784a^23 - 6275/3702784a^21 - 41173/1851392a^19 - 300629/3702784a^17 - 203665/925696a^15 - 127979/231424a^13 - 4592903/3702784a^11 - 2190645/925696a^9 - 901303/231424a^7 - 178345/28928a^5 - 27203/3616a^3 - 17011/3616a - 1, 153/1851392a^23 + 1909/1851392a^21 + 8399/925696a^19 + 65259/1851392a^17 + 45373/462848a^15 + 26565/115712a^13 + 957129/1851392a^11 + 466549/462848a^9 + 196165/115712a^7 + 18043/7232a^5 + 12329/3616a^3 + 5231/1808a - 1, 5449/3702784a^23 + 23101/3702784a^21 + 32779/1851392a^19 + 168491/3702784a^17 + 99647/925696a^15 + 50037/231424a^13 + 1426425/3702784a^11 + 588091/925696a^9 + 207113/231424a^7 + 27215/28928a^5 + 1953/3616a^3 + 1645/3616a - 1, 4061/1851392a^22 + 6929/1851392a^20 - 23841/925696a^18 - 231033/1851392a^16 - 164369/462848a^14 - 109527/115712a^12 - 4255411/1851392a^10 - 2104541/462848a^8 - 888171/115712a^6 - 182499/14464a^4 - 15377/904a^2 - 19987/1808, 1387/462848a^22 + 11867/462848a^20 + 23931/231424a^18 + 138905/462848a^16 + 21637/28928a^14 + 49191/28928a^12 + 1544123/462848a^10 + 333955/57856a^8 + 264025/28928a^6 + 86545/7232a^4 + 19319/1808a^2 + 380/113, 7085/3702784a^23 + 4613/1851392a^22 + 79969/3702784a^21 + 31081/1851392a^20 + 175983/1851392a^19 + 51679/925696a^18 + 1039223/3702784a^17 + 245407/1851392a^16 + 661903/925696a^15 + 148339/462848a^14 + 380745/231424a^13 + 76609/115712a^12 + 12334109/3702784a^11 + 2173941/1851392a^10 + 5423843/925696a^9 + 828191/462848a^8 + 2139957/231424a^7 + 277925/115712a^6 + 369765/28928a^5 + 36875/14464a^4 + 22863/1808a^3 - 455/1808a^2 + 20973/3616a - 5271/1808, 12637/1851392a^23 + 14655/1851392a^22 + 101265/1851392a^21 + 96795/1851392a^20 + 207199/925696a^19 + 167589/925696a^18 + 1219847/1851392a^17 + 907085/1851392a^16 + 774191/462848a^15 + 557885/462848a^14 + 440441/115712a^13 + 302803/115712a^12 + 14059725/1851392a^11 + 9167631/1851392a^10 + 6174435/462848a^9 + 3821289/462848a^8 + 2427781/115712a^7 + 1445767/115712a^6 + 410841/14464a^5 + 218635/14464a^4 + 6221/226a^3 + 4667/452a^2 + 24333/1808a + 3799/1808, 1745/1851392a^23 - 15491/1851392a^22 + 14901/1851392a^21 - 88815/1851392a^20 + 22731/925696a^19 - 148689/925696a^18 + 116067/1851392a^17 - 830169/1851392a^16 + 69683/462848a^15 - 509193/462848a^14 + 36229/115712a^13 - 276231/115712a^12 + 1015809/1851392a^11 - 8420115/1851392a^10 + 398855/462848a^9 - 3581189/462848a^8 + 132065/115712a^7 - 1374955/115712a^6 + 17253/14464a^5 - 208975/14464a^4 - 307/904a^3 - 5269/452a^2 - 2247/1808a - 12523/1808, 11901/1851392a^23 + 8877/925696a^22 + 78705/1851392a^21 + 66233/925696a^20 + 147487/925696a^19 + 137051/462848a^18 + 849767/1851392a^17 + 814375/925696a^16 + 530815/462848a^15 + 514753/231424a^14 + 297425/115712a^13 + 293505/57856a^12 + 9325037/1851392a^11 + 9441437/925696a^10 + 4069715/462848a^9 + 4153689/231424a^8 + 1603117/115712a^7 + 1645425/57856a^6 + 263133/14464a^5 + 140025/3616a^4 + 7691/452a^3 + 70045/1808a^2 + 17669/1808a + 18067/904 (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:	$ 18217606.15517919 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 18217606.15517919 \cdot 4}{28\cdot\sqrt{1041229780068396944496497143054336}}\cr\approx \mathstrut & 0.305336271977985 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 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 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 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 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 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 + 9*x^22 + 42*x^20 + 139*x^18 + 376*x^16 + 896*x^14 + 1905*x^12 + 3584*x^10 + 6016*x^8 + 8896*x^6 + 10752*x^4 + 9216*x^2 + 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

$C_2^3\times A_4$ (as 24T135):

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 96

The 32 conjugacy class representatives for $C_2^3\times A_4$

Character table for $C_2^3\times A_4$

Intermediate fields

$\Q(\sqrt{7}) $, $\Q(\sqrt{-1}) $, $\Q(\sqrt{-7}) $, $\Q(\zeta_{7})^+$, $\Q(i, \sqrt{7})$, 6.0.400967.1, 6.0.179633216.1, 6.6.25661888.1, 6.6.2806769.1, $\Q(\zeta_{28})^+$, 6.0.153664.1, $\Q(\zeta_{7})$, $\Q(\zeta_{28})$, 12.0.32268092290502656.2, 12.12.32268092290502656.1, 12.0.658532495724544.3, 12.0.32268092290502656.1, 12.0.32268092290502656.3, 12.0.7877952219361.1

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 sibling:	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	${\href{/padicField/3.6.0.1}{6} }^{4}$	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{12}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.6.0.1}{6} }^{4}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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
$2$ 2	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$7$ 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	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$167$ 167	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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