Normalized defining polynomial
\( x^{28} - 2 x^{27} + 45 x^{26} - 80 x^{25} + 1187 x^{24} - 1934 x^{23} + 22138 x^{22} - 33046 x^{21} + 318531 x^{20} - 435512 x^{19} + 3676840 x^{18} - 4581920 x^{17} + 34769671 x^{16} - 39193702 x^{15} + 271774836 x^{14} - 273990802 x^{13} + 1756548849 x^{12} - 1558280808 x^{11} + 9314576819 x^{10} - 7101150838 x^{9} + 39824889399 x^{8} - 25169366408 x^{7} + 133052209774 x^{6} - 65705100948 x^{5} + 328509907687 x^{4} - 113376601034 x^{3} + 537399587373 x^{2} - 97937579416 x + 440719107401 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 14]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(172491573797176117219488267129250761475686400000000000000=2^{28}\cdot 5^{14}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $101.97$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(580=2^{2}\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(579,·)$, $\chi_{580}(199,·)$, $\chi_{580}(521,·)$, $\chi_{580}(139,·)$, $\chi_{580}(499,·)$, $\chi_{580}(141,·)$, $\chi_{580}(399,·)$, $\chi_{580}(401,·)$, $\chi_{580}(339,·)$, $\chi_{580}(341,·)$, $\chi_{580}(121,·)$, $\chi_{580}(281,·)$, $\chi_{580}(219,·)$, $\chi_{580}(459,·)$, $\chi_{580}(161,·)$, $\chi_{580}(419,·)$, $\chi_{580}(81,·)$, $\chi_{580}(361,·)$, $\chi_{580}(299,·)$, $\chi_{580}(239,·)$, $\chi_{580}(241,·)$, $\chi_{580}(179,·)$, $\chi_{580}(181,·)$, $\chi_{580}(439,·)$, $\chi_{580}(441,·)$, $\chi_{580}(59,·)$, $\chi_{580}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{27} - \frac{25905638298912034746551197219971288637682043087278749242859827929848680878749586049055401204703841446161884999703349646465212}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{26} + \frac{86583621135012590672833272921841710813878971767807091191311333083291110101556182911841369511568913029024611972767923658804778}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{25} - \frac{23202965951253497874335590790764127409471501106532325271857845488625133825703223167338492584126349016443483443916401537818681}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{24} + \frac{47805516142306061586112849049600558974148065143548055066356624333009040382504165283481178675074592298074920815502605872641141}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{23} + \frac{433140954593672733918639383548167945627727559856433618661099149209657525246845199154271649979267621438195909903706529514804}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{22} - \frac{6653389142842487675004189464505125873483439395329444110623468279690342042762690399613231645234368567026971362362842774947638}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{21} - \frac{20051996387255421735875319627342250170841437670592238128884944116916320393989916176081687401697406401527197272466960557636690}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{20} + \frac{69850875666717891732968249652066926583050224014500464093557165708906028963544714265272958692922570197302640208540388998371948}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{19} + \frac{1210756392424721119754373179454715112491629981416673544132240803375057717799281618824716627278578641166547397630448766410994}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{18} - \frac{23873124931134270251788507924719925314114934584242358797991394414107836577161240793020461746757128688101749879254771761925955}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{17} + \frac{77852704396862613797656577050957543371543261530690852760488609911966707473772637118186281414769743385719799066837547635080686}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{16} + \frac{11747518857363796507291927560294268127043255452903415545271810844004576422585038020196711472708070911927837681177299580942792}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{15} + \frac{55041581176369950716047107027122043143672128094288363715758729462260345614143162915549803371757446827500866131049034672486499}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{14} - \frac{64530131624063982924232580909918958867201798846667944455621971704546091815164663637680839632991236309701755465537228387358650}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{13} + \frac{54936203640497317981697769458318869759395162787374158874504813783312705609239134617810287945721903990458552776357486544311264}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{12} + \frac{61587519155716575710419739529668003048726538656142460690893067590257532131268110488186723802689529848305428422661851216244566}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{11} + \frac{47710678366462423198929385034321639096150455789647454214432470564472857884917363405314484741635474973862334990386388685957968}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{10} + \frac{6133657419553911991298469595908795551040239770592181298798760806595319657403119428164352162317549822900498933334667226898753}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{9} - \frac{49153585233683187522076259736112822516794099070421701429603963449896644357646736852629665951361305873805494973859903367132467}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{8} - \frac{88213075665924133340486559818736403245912255623167348976579132985406138861448426806910668078268456827923705537706268310855485}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{7} - \frac{72878866588091541643120088485253408780756730323934327220555511459460898765488945587742028290434164698910781549697788835938688}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{6} + \frac{11354776771520430222578451859496135794762017367797171682096443219846384848694020953702246642880377583445708601129282658818136}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{5} - \frac{54946033625061863766233705590886634152914298567226907870227204368975550672312098256116628276953273972928991479015542817871442}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{4} + \frac{2264205066317820630246469946312387225505833282784260933169138738370233463992737445489574643773926253385864480018009937045748}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{3} + \frac{20265188237793630670151033818822658745937066861662657726352902857954397170970070628108429231354655404175290308961046965729983}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a^{2} - \frac{49515158194040675437345447067504525825610015839774130018536865876835196121194625435176999134000634286146912701756203817878625}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909} a - \frac{7134736063317095459061045149361745326775632339593047457645158663882850435358250369814197218906675347466741582492273473657426}{177405229381163771992589390900276493234649688975184856250752522393326685863759655533701703524078526346235927530782552499514909}$
Class group and class number
Not computed
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{14}$ (as 28T2):
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.14.7.1 | $x^{14} - 250 x^{8} + 15625 x^{2} - 312500$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| 5.14.7.1 | $x^{14} - 250 x^{8} + 15625 x^{2} - 312500$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
| $29$ | 29.14.13.1 | $x^{14} - 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
| 29.14.13.1 | $x^{14} - 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ | |