Normalized defining polynomial
\( x^{16} + 2764 x^{14} + 16284406 x^{12} + 22063918452 x^{10} + 41013254839662 x^{8} + 50513141676288564 x^{6} + 31250279689754897394 x^{4} + 9738852838043305012092 x^{2} + 1405380235130639218042893 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(160390192004889035274309827110029156806685344432046320325230592=2^{48}\cdot 3^{11}\cdot 7^{5}\cdot 294337^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $7723.67$ | 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: | $2, 3, 7, 294337$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{42} a^{8} - \frac{4}{21} a^{6} - \frac{1}{21} a^{4} - \frac{2}{7} a^{2} - \frac{1}{2}$, $\frac{1}{294} a^{9} + \frac{59}{147} a^{7} + \frac{20}{147} a^{5} - \frac{16}{49} a^{3} + \frac{1}{14} a$, $\frac{1}{14406} a^{10} + \frac{10}{7203} a^{8} - \frac{617}{7203} a^{6} - \frac{1567}{7203} a^{4} + \frac{85}{686} a^{2}$, $\frac{1}{302526} a^{11} - \frac{323}{302526} a^{9} + \frac{43973}{151263} a^{7} - \frac{7339}{16807} a^{5} - \frac{5207}{14406} a^{3} - \frac{1}{14} a$, $\frac{1}{14823774} a^{12} - \frac{323}{14823774} a^{10} - \frac{6448}{7411887} a^{8} + \frac{1070438}{2470629} a^{6} - \frac{82039}{705894} a^{4} + \frac{111}{686} a^{2}$, $\frac{1}{140188430718} a^{13} - \frac{9241}{140188430718} a^{11} - \frac{15860594}{70094215359} a^{9} + \frac{28507536884}{70094215359} a^{7} - \frac{1020011379}{2225213186} a^{5} - \frac{8256809}{19462506} a^{3} + \frac{3}{9457} a$, $\frac{1}{732221615765497508578793083634078393433983429934475117232831981298} a^{14} + \frac{3230248599451096006121440999911055975136094951824447083351}{732221615765497508578793083634078393433983429934475117232831981298} a^{12} - \frac{11782890405390436846162428046886576117370859807897038533370271}{366110807882748754289396541817039196716991714967237558616415990649} a^{10} + \frac{906677035237293772448044745091205670890729022096564917803172745}{122036935960916251429798847272346398905663904989079186205471996883} a^{8} + \frac{8115274039319744287542028512108834260503267075162720922008460327}{34867695988833214694228242077813256830189687139736910344420570538} a^{6} + \frac{5187355277150557447870119169959620232527924283298238908476755}{11295010038494724552714040193655088056426850385402303318568374} a^{4} + \frac{5853206215439065421962933962233953152049474620809577408878}{49395087631901127781548280730270064386706342793304533463127} a^{2} + \frac{5139133555271208561982034836779301582584107075991604}{35531458633673165672584811546045555562777414126745213}$, $\frac{1}{35878859172509377920360861098069841278265188066789280744408767083602} a^{15} - \frac{48440827773712777146864084224551901103409664794923958378}{17939429586254688960180430549034920639132594033394640372204383541801} a^{13} + \frac{7180229875660464047471722191043084267684151258755423029248245}{35878859172509377920360861098069841278265188066789280744408767083602} a^{11} + \frac{19559680849156660911109614865061209595730855937200593858680782593}{11959619724169792640120287032689947092755062688929760248136255694534} a^{9} + \frac{269676475039461570967559186195198855114272591899471449838778983115}{1708517103452827520017183861812849584679294669847108606876607956362} a^{7} + \frac{105017480202175247416862515078429559355743524973624552157200959}{276727745943120751541493984744549657382457834442356431304925163} a^{5} - \frac{420922511102812103648209693502186171706832216592545607186867}{1613572862642103507530577170522155436632407197914614759795482} a^{3} - \frac{117366463998173476989287316536170210724491868720404411323}{672042008597294255531269125581905637914372010793258958682} a$
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{213413765655731547708433}{553953008116445197341562957702911421303577577} a^{14} + \frac{570432228564634040613308863}{1107906016232890394683125915405822842607155154} a^{12} + \frac{2853623801736137079398279614163}{553953008116445197341562957702911421303577577} a^{10} + \frac{231458505551358170825276142399574}{553953008116445197341562957702911421303577577} a^{8} + \frac{252213434671253475383297007127844203}{26378714672211676063883950366805305776360837} a^{6} + \frac{323234672372125929394248160132882555}{153811747359834845853550731001780208608518} a^{4} - \frac{35686601084412117475390863622582799}{10676922626671862130608824864763307553} a^{2} - \frac{162682693900006575876234808987688571}{217896380136160451645078058464557297} \) (order $6$) | 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: | \( 4036710027520000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 29 conjugacy class representatives for t16n1022 |
| Character table for t16n1022 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.42384528.1, 8.0.2842620092391385389662208.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
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 | $16$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $16$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.8.6.2 | $x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 294337 | Data not computed | ||||||