Normalized defining polynomial
\( x^{28} - x^{27} - 81 x^{26} + 98 x^{25} + 2666 x^{24} - 3722 x^{23} - 46935 x^{22} + 73335 x^{21} + 486583 x^{20} - 833426 x^{19} - 3079355 x^{18} + 5705690 x^{17} + 11981008 x^{16} - 23840492 x^{15} - 28489796 x^{14} + 60403709 x^{13} + 41941965 x^{12} - 91572889 x^{11} - 41957881 x^{10} + 83155937 x^{9} + 31487031 x^{8} - 43876845 x^{7} - 17209192 x^{6} + 11875118 x^{5} + 5608146 x^{4} - 990770 x^{3} - 732425 x^{2} - 84867 x - 2789 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[28, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1407829094312471215113334241722636006626629352569580078125=5^{21}\cdot 43^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $109.91$ | 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: | $5, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(215=5\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{215}(128,·)$, $\chi_{215}(1,·)$, $\chi_{215}(2,·)$, $\chi_{215}(4,·)$, $\chi_{215}(118,·)$, $\chi_{215}(8,·)$, $\chi_{215}(137,·)$, $\chi_{215}(11,·)$, $\chi_{215}(64,·)$, $\chi_{215}(16,·)$, $\chi_{215}(82,·)$, $\chi_{215}(84,·)$, $\chi_{215}(21,·)$, $\chi_{215}(22,·)$, $\chi_{215}(88,·)$, $\chi_{215}(27,·)$, $\chi_{215}(32,·)$, $\chi_{215}(164,·)$, $\chi_{215}(113,·)$, $\chi_{215}(168,·)$, $\chi_{215}(41,·)$, $\chi_{215}(42,·)$, $\chi_{215}(44,·)$, $\chi_{215}(176,·)$, $\chi_{215}(108,·)$, $\chi_{215}(54,·)$, $\chi_{215}(121,·)$, $\chi_{215}(59,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{4909} a^{26} - \frac{2068}{4909} a^{25} + \frac{847}{4909} a^{24} + \frac{415}{4909} a^{23} + \frac{168}{4909} a^{22} - \frac{176}{4909} a^{21} - \frac{1028}{4909} a^{20} + \frac{527}{4909} a^{19} + \frac{751}{4909} a^{18} - \frac{1815}{4909} a^{17} - \frac{529}{4909} a^{16} - \frac{2277}{4909} a^{15} - \frac{500}{4909} a^{14} - \frac{614}{4909} a^{13} - \frac{318}{4909} a^{12} - \frac{2214}{4909} a^{11} + \frac{1854}{4909} a^{10} + \frac{922}{4909} a^{9} - \frac{419}{4909} a^{8} + \frac{1944}{4909} a^{7} + \frac{2039}{4909} a^{6} + \frac{1875}{4909} a^{5} - \frac{1526}{4909} a^{4} - \frac{527}{4909} a^{3} + \frac{497}{4909} a^{2} + \frac{1376}{4909} a + \frac{685}{4909}$, $\frac{1}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{27} - \frac{178250504533671837709504137722859146993931017501743981028303142255742122792}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{26} + \frac{534181213426548313090013153653604339227698606955762050589343483057277812517228}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{25} - \frac{300986939562654283578513824593893782717518597886802591986487109469311224839029}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{24} + \frac{979941955814695164808251945894221705477171755200425295202625299966047595090128}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{23} + \frac{224412586851889629268967691981266264963068358436958016788050625728360250862318}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{22} + \frac{430814372416696071614318413100722346758176390215928523344157593752393975652462}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{21} - \frac{418660941176816793742523994119788647945193224948744899364514281086869459874871}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{20} + \frac{343918568410205834778529547777468559584281885535865324953302838219814334928103}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{19} + \frac{782136340597875027526731084615906485380634728596484748964677228217160232486260}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{18} - \frac{856727707012204165113842727675905599561715861695254653372955346573065061485939}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{17} + \frac{660671221611598987351944282798683681191263633741338214333847874048893305582648}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{16} + \frac{455160638171690254210774138348587344588728583434302264903724680472968326837741}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{15} - \frac{90610167601967416352711672588443868393514827122330316071963521734842705823981}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{14} + \frac{555932762317086547658793954765074373990958207023547037395291296267699955755601}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{13} - \frac{199164162618316614758918885034829242354508309155976197268680908403351937887739}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{12} - \frac{145264376012862702635613265250024221039261880271882100001960998158745928263534}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{11} + \frac{747362586746318882279872921134514689278412663908459323774318188346376009407243}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{10} - \frac{310921816276285169253505947341846456085193862127893741363495384227629374209163}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{9} + \frac{246245307842456289793364116692894396317847439419234272220320706532510328914107}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{8} + \frac{737495521414870850276387267842167742964613959482855744157946664284520178325850}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{7} + \frac{532797428130991501300413691878775248353882846933258162650950532915685167367648}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{6} - \frac{870617789139422205031993958297180608731094774347611277690560214061888512781668}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{5} + \frac{249878072939834763641850541801095400697350769587695591009769197139301068622904}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{4} + \frac{381937979882263252673234807477956683087912677137361477578200420338951159494586}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{3} + \frac{947123760836823857695614415197500753773894007400716351993878641802554839371366}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a^{2} - \frac{159297108187883193358964127714274569382299015321215826274649098144285022441265}{2104023369815633366612853528833610649076543281388238374746612688940071157871109} a - \frac{110069391305116738138877936850985627307622698700821676627596381128186857656}{754400634569965351958714065555256597015612506772405297506852882373636126881}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $27$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 34667735947885390000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.231125.1, 7.7.6321363049.1, 14.14.3121846156036138781328125.1 |
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 | $28$ | $28$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ | $28$ | $28$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ | $28$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ | R | $28$ | $28$ | ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 43 | Data not computed | ||||||