Normalized defining polynomial
\( x^{24} - x^{23} + 14 x^{22} - 19 x^{21} - 11 x^{20} - 88 x^{19} - 536 x^{18} + 80 x^{17} - 139 x^{16} - 921 x^{15} + 8445 x^{14} + 3107 x^{13} + 29288 x^{12} + 72619 x^{11} + 63829 x^{10} + 241364 x^{9} + 226913 x^{8} + 188822 x^{7} + 582221 x^{6} - 247054 x^{5} + 681404 x^{4} - 328603 x^{3} + 359338 x^{2} - 70083 x + 149227 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 12]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5513001989803802379160200522041204586631217=7^{20}\cdot 17^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.38$ | 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: | $7, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(119=7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(66,·)$, $\chi_{119}(67,·)$, $\chi_{119}(4,·)$, $\chi_{119}(72,·)$, $\chi_{119}(76,·)$, $\chi_{119}(16,·)$, $\chi_{119}(81,·)$, $\chi_{119}(18,·)$, $\chi_{119}(19,·)$, $\chi_{119}(86,·)$, $\chi_{119}(87,·)$, $\chi_{119}(26,·)$, $\chi_{119}(30,·)$, $\chi_{119}(104,·)$, $\chi_{119}(106,·)$, $\chi_{119}(110,·)$, $\chi_{119}(111,·)$, $\chi_{119}(50,·)$, $\chi_{119}(83,·)$, $\chi_{119}(94,·)$, $\chi_{119}(59,·)$, $\chi_{119}(117,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{4}$, $\frac{1}{13} a^{17} - \frac{1}{13} a^{5}$, $\frac{1}{13} a^{18} - \frac{1}{13} a^{6}$, $\frac{1}{13} a^{19} - \frac{1}{13} a^{7}$, $\frac{1}{13} a^{20} - \frac{1}{13} a^{8}$, $\frac{1}{42419} a^{21} + \frac{449}{42419} a^{20} + \frac{157}{42419} a^{19} - \frac{203}{42419} a^{18} + \frac{1321}{42419} a^{17} + \frac{61}{3263} a^{16} - \frac{101}{42419} a^{15} - \frac{1325}{42419} a^{14} - \frac{113}{3263} a^{13} + \frac{1486}{3263} a^{12} + \frac{39}{251} a^{11} - \frac{1212}{3263} a^{10} + \frac{3717}{42419} a^{9} - \frac{14879}{42419} a^{8} + \frac{6525}{42419} a^{7} - \frac{7922}{42419} a^{6} + \frac{13200}{42419} a^{5} + \frac{1553}{3263} a^{4} + \frac{16845}{42419} a^{3} - \frac{17902}{42419} a^{2} - \frac{544}{3263} a + \frac{11}{251}$, $\frac{1}{1312995307} a^{22} - \frac{248}{100999639} a^{21} - \frac{13523077}{1312995307} a^{20} + \frac{38677026}{1312995307} a^{19} - \frac{25924822}{1312995307} a^{18} - \frac{50242833}{1312995307} a^{17} - \frac{6051796}{1312995307} a^{16} + \frac{35936348}{1312995307} a^{15} + \frac{5547223}{1312995307} a^{14} - \frac{1358788}{100999639} a^{13} + \frac{5460427}{100999639} a^{12} + \frac{5112871}{100999639} a^{11} + \frac{474133152}{1312995307} a^{10} - \frac{1465741}{100999639} a^{9} - \frac{225778180}{1312995307} a^{8} - \frac{63185718}{1312995307} a^{7} - \frac{298628277}{1312995307} a^{6} - \frac{24875184}{1312995307} a^{5} + \frac{379449015}{1312995307} a^{4} - \frac{653654215}{1312995307} a^{3} + \frac{14077369}{1312995307} a^{2} - \frac{31316724}{100999639} a + \frac{2518291}{7769203}$, $\frac{1}{2819021739358092823521975350544739364944470472619033} a^{23} + \frac{960875739758227650990661785269134222471214}{2819021739358092823521975350544739364944470472619033} a^{22} - \frac{19921339646139828832069000411700915137200501376}{2819021739358092823521975350544739364944470472619033} a^{21} + \frac{50169703350951425994849227539444280344637278460650}{2819021739358092823521975350544739364944470472619033} a^{20} + \frac{1471918564951017924850088029380700050089322307848}{216847826104468678732459642349595335764959267124541} a^{19} - \frac{60101644184862834416627304860310000890956382056176}{2819021739358092823521975350544739364944470472619033} a^{18} + \frac{74548432544193314542296801571303170165591785732112}{2819021739358092823521975350544739364944470472619033} a^{17} - \frac{52298887428502624794220558152128210107827496297717}{2819021739358092823521975350544739364944470472619033} a^{16} - \frac{77128628735133718875873490518543723270754341042104}{2819021739358092823521975350544739364944470472619033} a^{15} - \frac{40276631884629446720129032068307724176075669801358}{2819021739358092823521975350544739364944470472619033} a^{14} - \frac{3949567317779525785951049656839272352695929198222}{216847826104468678732459642349595335764959267124541} a^{13} + \frac{48127244727613732675592613944234658087601841093623}{216847826104468678732459642349595335764959267124541} a^{12} - \frac{626755586082942046542295784675874185405109138169150}{2819021739358092823521975350544739364944470472619033} a^{11} - \frac{107268836341164198141109544765540539180750381546902}{2819021739358092823521975350544739364944470472619033} a^{10} + \frac{142123692253507888947758690961024517057270375830447}{2819021739358092823521975350544739364944470472619033} a^{9} + \frac{179267313153201713437955841483555025089467961181231}{2819021739358092823521975350544739364944470472619033} a^{8} + \frac{8808285786106915108121387376708595876082407401061}{216847826104468678732459642349595335764959267124541} a^{7} - \frac{70921664562653516635961863619198828613985652975384}{2819021739358092823521975350544739364944470472619033} a^{6} + \frac{684305301282980020663014320262508916640476164665508}{2819021739358092823521975350544739364944470472619033} a^{5} + \frac{1364661260151643669160948244483855482381834606104107}{2819021739358092823521975350544739364944470472619033} a^{4} + \frac{191314580854276635696967777310511882172047355231536}{2819021739358092823521975350544739364944470472619033} a^{3} + \frac{509040857127354072998343209830656967914244558779534}{2819021739358092823521975350544739364944470472619033} a^{2} - \frac{2275809160907904740788179199162093605168374977366}{216847826104468678732459642349595335764959267124541} a - \frac{7944250836078000336821242492381224894970771776581}{16680602008036052210189203257661179674227635932657}$
Class group and class number
$C_{45}\times C_{90}$, which has order $4050$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 250243842.68845215 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 24 |
| The 24 conjugacy class representatives for $C_{24}$ |
| Character table for $C_{24}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.0.985223153873.1, 12.12.683635509017782097.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | $24$ | $24$ | R | $24$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ | R | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 17 | Data not computed | ||||||