Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256)
gp: K = bnfinit(y^24 + 30*y^22 + 423*y^20 + 3156*y^18 + 11967*y^16 + 32862*y^14 + 196601*y^12 + 135120*y^10 + 1571520*y^8 + 770688*y^6 + 4961280*y^4 + 3182592*y^2 + 3936256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256)
\( x^{24} + 30 x^{22} + 423 x^{20} + 3156 x^{18} + 11967 x^{16} + 32862 x^{14} + 196601 x^{12} + \cdots + 3936256 \)
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: |
\(79918740480958990570269904896000000000000\)
\(\medspace = 2^{24}\cdot 3^{28}\cdot 5^{12}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.61\) | 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: | $2\cdot 3^{7/6}5^{1/2}31^{1/2}\approx 89.70926709273317$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(31\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{9}-\frac{1}{16}a^{7}-\frac{3}{32}a^{5}$, $\frac{1}{32}a^{10}+\frac{1}{32}a^{6}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{7}+\frac{1}{16}a^{5}$, $\frac{1}{256}a^{12}-\frac{1}{256}a^{10}-\frac{5}{256}a^{8}+\frac{29}{256}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}$, $\frac{1}{256}a^{13}-\frac{1}{256}a^{11}+\frac{3}{256}a^{9}+\frac{13}{256}a^{7}+\frac{1}{32}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{256}a^{14}+\frac{1}{128}a^{10}-\frac{1}{32}a^{8}+\frac{5}{256}a^{6}+\frac{1}{16}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{512}a^{15}-\frac{3}{256}a^{11}-\frac{1}{64}a^{9}-\frac{3}{512}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a$, $\frac{1}{1024}a^{16}-\frac{1}{512}a^{12}-\frac{3}{256}a^{10}-\frac{23}{1024}a^{8}+\frac{13}{256}a^{6}-\frac{1}{8}a^{4}+\frac{1}{16}a^{2}+\frac{1}{4}$, $\frac{1}{2048}a^{17}-\frac{1}{1024}a^{13}+\frac{5}{512}a^{11}-\frac{23}{2048}a^{9}+\frac{21}{512}a^{7}-\frac{1}{32}a^{5}+\frac{1}{32}a^{3}+\frac{1}{8}a$, $\frac{1}{2048}a^{18}-\frac{1}{1024}a^{14}-\frac{1}{512}a^{12}+\frac{1}{2048}a^{10}-\frac{13}{512}a^{8}-\frac{31}{256}a^{6}+\frac{1}{32}a^{4}+\frac{3}{8}a^{2}+\frac{1}{4}$, $\frac{1}{63488}a^{19}+\frac{7}{31744}a^{17}+\frac{21}{31744}a^{15}-\frac{3}{7936}a^{13}-\frac{567}{63488}a^{11}+\frac{97}{31744}a^{9}-\frac{175}{15872}a^{7}+\frac{29}{248}a^{5}-\frac{65}{496}a^{3}+\frac{109}{248}a$, $\frac{1}{82534400}a^{20}+\frac{5253}{82534400}a^{18}+\frac{13723}{41267200}a^{16}+\frac{2801}{3174400}a^{14}+\frac{155549}{82534400}a^{12}-\frac{591007}{82534400}a^{10}-\frac{27641}{20633600}a^{8}+\frac{82249}{1031680}a^{6}+\frac{11057}{99200}a^{4}-\frac{997}{4960}a^{2}-\frac{981}{2600}$, $\frac{1}{165068800}a^{21}-\frac{1247}{165068800}a^{19}+\frac{8523}{82534400}a^{17}-\frac{1499}{6348800}a^{15}-\frac{172051}{165068800}a^{13}+\frac{1482493}{165068800}a^{11}-\frac{2611}{1331200}a^{9}-\frac{124841}{2063360}a^{7}+\frac{5757}{198400}a^{5}+\frac{1943}{9920}a^{3}-\frac{36261}{161200}a$, $\frac{1}{16\!\cdots\!00}a^{22}+\frac{52208048989}{16\!\cdots\!00}a^{20}+\frac{99079472889277}{83\!\cdots\!00}a^{18}-\frac{31\!\cdots\!59}{83\!\cdots\!00}a^{16}-\frac{28\!\cdots\!03}{33\!\cdots\!00}a^{14}+\frac{29\!\cdots\!57}{16\!\cdots\!00}a^{12}-\frac{410980679134543}{32\!\cdots\!00}a^{10}+\frac{25\!\cdots\!77}{80\!\cdots\!00}a^{8}+\frac{183870148554771}{26\!\cdots\!00}a^{6}-\frac{77952258160309}{630496851689000}a^{4}-\frac{12\!\cdots\!31}{16\!\cdots\!00}a^{2}-\frac{20470889000227}{66100476386750}$, $\frac{1}{16\!\cdots\!00}a^{23}-\frac{24742495803}{83\!\cdots\!00}a^{21}+\frac{60568261853519}{16\!\cdots\!00}a^{19}-\frac{220692939441493}{10\!\cdots\!00}a^{17}+\frac{23\!\cdots\!03}{33\!\cdots\!00}a^{15}+\frac{51\!\cdots\!01}{83\!\cdots\!00}a^{13}-\frac{18\!\cdots\!27}{12\!\cdots\!00}a^{11}+\frac{12\!\cdots\!49}{41\!\cdots\!00}a^{9}-\frac{18\!\cdots\!91}{10\!\cdots\!00}a^{7}+\frac{14\!\cdots\!97}{20\!\cdots\!00}a^{5}+\frac{839504820460901}{65\!\cdots\!00}a^{3}+\frac{37\!\cdots\!49}{16\!\cdots\!00}a$
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
$C_{12}\times C_{24}$, which has order $288$ (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: |
\( \frac{891711267}{13124588099584000} a^{23} + \frac{4321222837}{3281147024896000} a^{21} + \frac{107832578523}{13124588099584000} a^{19} - \frac{178054120549}{3281147024896000} a^{17} - \frac{2384137233019}{2624917619916800} a^{15} - \frac{5913408156779}{3281147024896000} a^{13} + \frac{9825095482941}{1009583699968000} a^{11} - \frac{267720061576117}{3281147024896000} a^{9} + \frac{194724084767253}{820286756224000} a^{7} - \frac{9151283329351}{15774745312000} a^{5} + \frac{37862075235567}{51267922264000} a^{3} - \frac{3260576856167}{12816980566000} a \)
(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: |
$\frac{268055964477}{67\!\cdots\!40}a^{23}+\frac{3947746979619}{33\!\cdots\!20}a^{21}+\frac{108960275290743}{67\!\cdots\!40}a^{19}+\frac{7538743420491}{64\!\cdots\!60}a^{17}+\frac{551837832782703}{13\!\cdots\!88}a^{15}+\frac{36\!\cdots\!07}{33\!\cdots\!20}a^{13}+\frac{49\!\cdots\!53}{67\!\cdots\!40}a^{11}+\frac{19\!\cdots\!53}{16\!\cdots\!60}a^{9}+\frac{18\!\cdots\!21}{32\!\cdots\!80}a^{7}+\frac{180947230976049}{807035970161920}a^{5}+\frac{33\!\cdots\!37}{26\!\cdots\!40}a^{3}+\frac{80289063640101}{50439748135120}a$, $\frac{891711267}{13\!\cdots\!00}a^{23}+\frac{4321222837}{32\!\cdots\!00}a^{21}+\frac{107832578523}{13\!\cdots\!00}a^{19}-\frac{178054120549}{32\!\cdots\!00}a^{17}-\frac{2384137233019}{26\!\cdots\!00}a^{15}-\frac{5913408156779}{32\!\cdots\!00}a^{13}+\frac{9825095482941}{10\!\cdots\!00}a^{11}-\frac{267720061576117}{32\!\cdots\!00}a^{9}+\frac{194724084767253}{820286756224000}a^{7}-\frac{9151283329351}{15774745312000}a^{5}+\frac{37862075235567}{51267922264000}a^{3}-\frac{3260576856167}{12816980566000}a-1$, $\frac{2357142796167}{41\!\cdots\!00}a^{23}+\frac{23608633661}{55\!\cdots\!00}a^{22}+\frac{10506457516367}{64\!\cdots\!00}a^{21}+\frac{579947177669}{55\!\cdots\!00}a^{20}+\frac{18\!\cdots\!21}{83\!\cdots\!00}a^{19}+\frac{3019374024007}{27\!\cdots\!00}a^{18}+\frac{63\!\cdots\!79}{41\!\cdots\!00}a^{17}+\frac{689934758417}{21\!\cdots\!00}a^{16}+\frac{952513594836789}{20\!\cdots\!00}a^{15}-\frac{5748376451839}{22\!\cdots\!20}a^{14}+\frac{76\!\cdots\!43}{83\!\cdots\!00}a^{13}-\frac{25097687487473}{17\!\cdots\!00}a^{12}+\frac{71\!\cdots\!41}{83\!\cdots\!00}a^{11}+\frac{361725649840211}{13\!\cdots\!00}a^{10}-\frac{22\!\cdots\!21}{52\!\cdots\!00}a^{9}-\frac{272472184972681}{868498974512000}a^{8}+\frac{37\!\cdots\!31}{52\!\cdots\!00}a^{7}+\frac{34426800288099}{133615226848000}a^{6}-\frac{23\!\cdots\!27}{10\!\cdots\!00}a^{5}-\frac{30345606289097}{16701903356000}a^{4}+\frac{400378704351941}{264401905547000}a^{3}-\frac{4095737956429}{13570296476750}a^{2}+\frac{34\!\cdots\!91}{81\!\cdots\!00}a-\frac{174801486241}{67346384500}$, $\frac{1074223649}{10\!\cdots\!00}a^{22}+\frac{32274924821}{10\!\cdots\!00}a^{20}+\frac{446390847051}{10\!\cdots\!00}a^{18}+\frac{1565354474989}{528005415808000}a^{16}+\frac{366933250859}{42240433264640}a^{14}+\frac{9212026927933}{10\!\cdots\!00}a^{12}+\frac{133282630741521}{10\!\cdots\!00}a^{10}+\frac{9982291479611}{264002707904000}a^{8}+\frac{49594365197557}{132001353952000}a^{6}+\frac{1159080590033}{1269243788000}a^{4}-\frac{5709742896951}{4125042311000}a^{2}+\frac{52105554687}{66532940500}$, $\frac{11423398965071}{16\!\cdots\!00}a^{23}-\frac{10543714056191}{16\!\cdots\!00}a^{22}+\frac{5215112629501}{26\!\cdots\!00}a^{21}-\frac{233969250353809}{16\!\cdots\!00}a^{20}+\frac{44\!\cdots\!19}{16\!\cdots\!00}a^{19}-\frac{483624266258511}{41\!\cdots\!00}a^{18}+\frac{72\!\cdots\!23}{41\!\cdots\!00}a^{17}+\frac{13\!\cdots\!39}{83\!\cdots\!00}a^{16}+\frac{13\!\cdots\!81}{33\!\cdots\!00}a^{15}+\frac{28\!\cdots\!41}{33\!\cdots\!00}a^{14}-\frac{323194280147889}{52\!\cdots\!00}a^{13}+\frac{59\!\cdots\!23}{16\!\cdots\!00}a^{12}+\frac{12\!\cdots\!49}{16\!\cdots\!00}a^{11}-\frac{19\!\cdots\!87}{83\!\cdots\!00}a^{10}-\frac{14\!\cdots\!11}{41\!\cdots\!00}a^{9}+\frac{41\!\cdots\!33}{67\!\cdots\!00}a^{8}+\frac{23\!\cdots\!89}{10\!\cdots\!00}a^{7}-\frac{25\!\cdots\!97}{52\!\cdots\!00}a^{6}+\frac{12\!\cdots\!17}{20\!\cdots\!00}a^{5}+\frac{31\!\cdots\!69}{10\!\cdots\!00}a^{4}-\frac{11\!\cdots\!79}{65\!\cdots\!00}a^{3}+\frac{25\!\cdots\!67}{32\!\cdots\!00}a^{2}+\frac{55\!\cdots\!89}{16\!\cdots\!00}a+\frac{173589359347783}{264401905547000}$, $\frac{426542349907}{20\!\cdots\!00}a^{23}+\frac{4260109895751}{41\!\cdots\!00}a^{22}+\frac{148886829771659}{16\!\cdots\!00}a^{21}+\frac{271771518259063}{83\!\cdots\!00}a^{20}+\frac{27\!\cdots\!99}{16\!\cdots\!00}a^{19}+\frac{40\!\cdots\!13}{83\!\cdots\!00}a^{18}+\frac{14\!\cdots\!21}{83\!\cdots\!00}a^{17}+\frac{15\!\cdots\!37}{41\!\cdots\!00}a^{16}+\frac{15\!\cdots\!51}{16\!\cdots\!00}a^{15}+\frac{28\!\cdots\!17}{20\!\cdots\!00}a^{14}+\frac{36\!\cdots\!67}{16\!\cdots\!00}a^{13}+\frac{18\!\cdots\!79}{83\!\cdots\!00}a^{12}+\frac{81\!\cdots\!79}{16\!\cdots\!00}a^{11}+\frac{12\!\cdots\!73}{83\!\cdots\!00}a^{10}+\frac{15\!\cdots\!99}{41\!\cdots\!00}a^{9}+\frac{60\!\cdots\!23}{20\!\cdots\!00}a^{8}-\frac{97\!\cdots\!21}{10\!\cdots\!00}a^{7}+\frac{13\!\cdots\!43}{52\!\cdots\!00}a^{6}+\frac{44\!\cdots\!47}{20\!\cdots\!00}a^{5}+\frac{24\!\cdots\!69}{10\!\cdots\!00}a^{4}-\frac{47\!\cdots\!69}{65\!\cdots\!00}a^{3}-\frac{25\!\cdots\!83}{10\!\cdots\!00}a^{2}-\frac{574122775578501}{16\!\cdots\!00}a+\frac{651262449505233}{264401905547000}$, $\frac{884725536133}{16\!\cdots\!00}a^{23}+\frac{83103867879}{16\!\cdots\!00}a^{22}+\frac{88261035281}{52\!\cdots\!00}a^{21}+\frac{468737565093}{16\!\cdots\!00}a^{20}-\frac{332528319227703}{16\!\cdots\!00}a^{19}-\frac{1209962012353}{10\!\cdots\!00}a^{18}-\frac{17\!\cdots\!41}{41\!\cdots\!00}a^{17}-\frac{38742820338767}{16\!\cdots\!60}a^{16}-\frac{11\!\cdots\!33}{33\!\cdots\!00}a^{15}-\frac{21\!\cdots\!73}{16\!\cdots\!00}a^{14}-\frac{10\!\cdots\!39}{10\!\cdots\!00}a^{13}+\frac{57\!\cdots\!41}{16\!\cdots\!00}a^{12}-\frac{33\!\cdots\!13}{16\!\cdots\!00}a^{11}+\frac{36\!\cdots\!01}{83\!\cdots\!00}a^{10}-\frac{65\!\cdots\!23}{41\!\cdots\!00}a^{9}-\frac{21\!\cdots\!47}{52\!\cdots\!00}a^{8}+\frac{11\!\cdots\!47}{10\!\cdots\!00}a^{7}+\frac{33\!\cdots\!63}{52\!\cdots\!00}a^{6}-\frac{20\!\cdots\!69}{20\!\cdots\!00}a^{5}-\frac{276745458709659}{10\!\cdots\!00}a^{4}-\frac{74\!\cdots\!67}{65\!\cdots\!00}a^{3}+\frac{324420077685467}{204911476798925}a^{2}-\frac{43\!\cdots\!73}{16\!\cdots\!00}a+\frac{33145629345507}{26440190554700}$, $\frac{5264532497439}{16\!\cdots\!00}a^{23}-\frac{1730655255333}{83\!\cdots\!00}a^{22}+\frac{66154904429863}{83\!\cdots\!00}a^{21}-\frac{37450905630597}{83\!\cdots\!00}a^{20}+\frac{14\!\cdots\!21}{16\!\cdots\!00}a^{19}-\frac{180747273130131}{41\!\cdots\!00}a^{18}+\frac{560985899016641}{20\!\cdots\!00}a^{17}-\frac{53130379292941}{32\!\cdots\!00}a^{16}-\frac{61\!\cdots\!31}{33\!\cdots\!00}a^{15}-\frac{11\!\cdots\!53}{16\!\cdots\!00}a^{14}-\frac{80\!\cdots\!41}{83\!\cdots\!00}a^{13}-\frac{12\!\cdots\!21}{83\!\cdots\!00}a^{12}+\frac{62\!\cdots\!91}{16\!\cdots\!00}a^{11}-\frac{21\!\cdots\!13}{20\!\cdots\!00}a^{10}-\frac{54\!\cdots\!99}{41\!\cdots\!00}a^{9}+\frac{67\!\cdots\!89}{10\!\cdots\!00}a^{8}+\frac{11\!\cdots\!51}{10\!\cdots\!00}a^{7}-\frac{25\!\cdots\!47}{20\!\cdots\!00}a^{6}+\frac{42\!\cdots\!03}{20\!\cdots\!00}a^{5}+\frac{611578608438221}{25\!\cdots\!00}a^{4}-\frac{95\!\cdots\!11}{65\!\cdots\!00}a^{3}-\frac{88\!\cdots\!79}{16\!\cdots\!00}a^{2}+\frac{93\!\cdots\!51}{16\!\cdots\!00}a-\frac{57965927237287}{10169304059500}$, $\frac{1274045133}{791808876385280}a^{23}+\frac{15040435687957}{83\!\cdots\!00}a^{22}+\frac{65466917959}{12\!\cdots\!00}a^{21}+\frac{460667378227213}{83\!\cdots\!00}a^{20}+\frac{13288752606971}{15\!\cdots\!00}a^{19}+\frac{32\!\cdots\!99}{41\!\cdots\!00}a^{18}+\frac{56902774647081}{79\!\cdots\!00}a^{17}+\frac{24\!\cdots\!57}{41\!\cdots\!00}a^{16}+\frac{261688674566901}{79\!\cdots\!00}a^{15}+\frac{34\!\cdots\!77}{16\!\cdots\!00}a^{14}+\frac{13\!\cdots\!43}{15\!\cdots\!00}a^{13}+\frac{10\!\cdots\!39}{27\!\cdots\!00}a^{12}+\frac{52\!\cdots\!51}{15\!\cdots\!00}a^{11}+\frac{56\!\cdots\!77}{20\!\cdots\!00}a^{10}+\frac{33\!\cdots\!33}{39\!\cdots\!00}a^{9}+\frac{11\!\cdots\!47}{52\!\cdots\!00}a^{8}+\frac{389728096221331}{197952219096320}a^{7}+\frac{24\!\cdots\!59}{16\!\cdots\!00}a^{6}+\frac{88480224118809}{19033867220800}a^{5}+\frac{54\!\cdots\!91}{25\!\cdots\!00}a^{4}+\frac{15037664511989}{2474402738704}a^{3}+\frac{57\!\cdots\!33}{81\!\cdots\!00}a^{2}+\frac{69802462003593}{15465017116900}a+\frac{117993074338787}{66100476386750}$, $\frac{6003700894263}{16\!\cdots\!00}a^{23}+\frac{7444087559301}{41\!\cdots\!00}a^{22}+\frac{6500122202861}{52\!\cdots\!00}a^{21}+\frac{47729027758121}{10\!\cdots\!00}a^{20}+\frac{33\!\cdots\!87}{16\!\cdots\!00}a^{19}+\frac{21\!\cdots\!39}{41\!\cdots\!00}a^{18}+\frac{75\!\cdots\!59}{41\!\cdots\!00}a^{17}+\frac{10\!\cdots\!69}{52\!\cdots\!00}a^{16}+\frac{11\!\cdots\!25}{13\!\cdots\!88}a^{15}-\frac{48\!\cdots\!69}{83\!\cdots\!00}a^{14}+\frac{12\!\cdots\!03}{52\!\cdots\!00}a^{13}-\frac{45\!\cdots\!47}{10\!\cdots\!00}a^{12}+\frac{11\!\cdots\!77}{16\!\cdots\!00}a^{11}+\frac{46\!\cdots\!13}{32\!\cdots\!00}a^{10}+\frac{85\!\cdots\!57}{41\!\cdots\!00}a^{9}-\frac{13\!\cdots\!27}{13\!\cdots\!00}a^{8}+\frac{57\!\cdots\!67}{10\!\cdots\!00}a^{7}+\frac{26\!\cdots\!09}{26\!\cdots\!00}a^{6}+\frac{15\!\cdots\!21}{20\!\cdots\!00}a^{5}-\frac{23\!\cdots\!23}{50\!\cdots\!00}a^{4}+\frac{86\!\cdots\!13}{65\!\cdots\!00}a^{3}-\frac{24\!\cdots\!14}{10\!\cdots\!25}a^{2}+\frac{951589980461007}{16\!\cdots\!00}a-\frac{722207668808761}{132200952773500}$, $\frac{3344297644227}{12\!\cdots\!00}a^{23}-\frac{110392103}{42\!\cdots\!00}a^{22}+\frac{631758329983447}{83\!\cdots\!00}a^{21}+\frac{2506249513}{42\!\cdots\!00}a^{20}+\frac{17\!\cdots\!69}{16\!\cdots\!00}a^{19}+\frac{60301513289}{21\!\cdots\!00}a^{18}+\frac{74\!\cdots\!57}{10\!\cdots\!00}a^{17}+\frac{948495764417}{21\!\cdots\!00}a^{16}+\frac{60\!\cdots\!61}{25\!\cdots\!00}a^{15}+\frac{562218193237}{168961733058560}a^{14}+\frac{47\!\cdots\!51}{83\!\cdots\!00}a^{13}+\frac{44651189044949}{42\!\cdots\!00}a^{12}+\frac{73\!\cdots\!99}{16\!\cdots\!00}a^{11}+\frac{12319185295811}{528005415808000}a^{10}-\frac{46\!\cdots\!01}{41\!\cdots\!00}a^{9}+\frac{61064824323127}{264002707904000}a^{8}+\frac{39\!\cdots\!59}{10\!\cdots\!00}a^{7}-\frac{19364764199151}{132001353952000}a^{6}-\frac{27\!\cdots\!53}{20\!\cdots\!00}a^{5}+\frac{159486307041}{79327736750}a^{4}+\frac{52\!\cdots\!27}{50\!\cdots\!00}a^{3}-\frac{4647585581757}{4125042311000}a^{2}+\frac{67\!\cdots\!99}{16\!\cdots\!00}a+\frac{331397184509}{66532940500}$
| 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: | \( 3654264020.3138604 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 3654264020.3138604 \cdot 288}{12\cdot\sqrt{79918740480958990570269904896000000000000}}\cr\approx \mathstrut & 1.17447903414768 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256)
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 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256, 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 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256);
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 + 30*x^22 + 423*x^20 + 3156*x^18 + 11967*x^16 + 32862*x^14 + 196601*x^12 + 135120*x^10 + 1571520*x^8 + 770688*x^6 + 4961280*x^4 + 3182592*x^2 + 3936256);
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
$C_2^2\times D_6$ (as 24T30):
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 48 |
| The 24 conjugacy class representatives for $C_2^2\times D_6$ |
| Character table for $C_2^2\times D_6$ |
Intermediate fields
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 |
| 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 | R | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.2.0.1}{2} }^{12}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ |
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| 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 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |