Normalized defining polynomial
\( x^{20} - 4 x^{19} + 96 x^{18} + 122 x^{17} + 4620 x^{16} + 18596 x^{15} + 184045 x^{14} + 728624 x^{13} + 6000753 x^{12} + 13556428 x^{11} + 139124895 x^{10} + 136683460 x^{9} + 2117576320 x^{8} - 289614572 x^{7} + 18977386826 x^{6} - 37574492902 x^{5} + 70623059201 x^{4} - 324133328144 x^{3} + 72454618799 x^{2} - 66338130030 x + 1082598671174 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(78148390643938933109518475030237312085224587264=2^{20}\cdot 17^{15}\cdot 53^{13}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $221.13$ | 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: | $2, 17, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{51} a^{16} - \frac{2}{51} a^{15} - \frac{20}{51} a^{14} + \frac{3}{17} a^{13} - \frac{8}{17} a^{12} + \frac{6}{17} a^{11} + \frac{1}{3} a^{10} - \frac{3}{17} a^{9} - \frac{8}{51} a^{8} + \frac{2}{17} a^{7} - \frac{1}{3} a^{6} + \frac{7}{17} a^{5} + \frac{19}{51} a^{4} - \frac{6}{17} a^{3} - \frac{1}{51} a^{2} + \frac{8}{17} a + \frac{1}{51}$, $\frac{1}{51} a^{17} - \frac{7}{51} a^{15} + \frac{20}{51} a^{14} + \frac{11}{51} a^{13} + \frac{4}{51} a^{12} + \frac{19}{51} a^{11} + \frac{8}{51} a^{10} + \frac{25}{51} a^{9} - \frac{10}{51} a^{8} + \frac{4}{17} a^{7} + \frac{7}{17} a^{6} - \frac{7}{51} a^{5} - \frac{14}{51} a^{4} + \frac{14}{51} a^{3} + \frac{22}{51} a^{2} - \frac{19}{51} a + \frac{19}{51}$, $\frac{1}{153} a^{18} - \frac{1}{153} a^{16} + \frac{25}{153} a^{15} + \frac{44}{153} a^{14} - \frac{3}{17} a^{13} + \frac{11}{153} a^{12} - \frac{71}{153} a^{11} - \frac{43}{153} a^{10} + \frac{38}{153} a^{9} - \frac{4}{17} a^{8} + \frac{74}{153} a^{7} - \frac{25}{51} a^{6} + \frac{44}{153} a^{5} - \frac{8}{153} a^{4} - \frac{35}{153} a^{3} - \frac{76}{153} a^{2} - \frac{58}{153} a + \frac{74}{153}$, $\frac{1}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{19} + \frac{4742868699701306431532316171507472453314549046876991028619153123211531715553182300805381130741065833996422635}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{18} - \frac{24529917317034847549436828270538830806473945949096692313255150615113257493137590401786669730282294877084774575}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{17} - \frac{928614395481280526375214758847648800145267509667447018954566055058089344971483795745380492442458377950112737}{835009510396320595361055059946475043484240810518376082708902847927683337754288729868862192181918943525785120604} a^{16} - \frac{19784945602698379643793070318424276016004207398220716948428241827127369794764978826812424455761884131489265}{143398507710170117698961885616774007124204157739717685507281959115178316633056625428277896648105605963555748} a^{15} - \frac{602191434968482665863047908384453802883198770238411684696331250565179100925718024328286343618858130513795317759}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{14} + \frac{145059334069419447519930654982465185426145593105952534345393835699247081134276939998062332261563384688274340076}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{13} - \frac{33772903099695737159721202870008290243415883881631582822240964535672418743832697543764457838593374122540269612}{69584125866360049613421254995539586957020067543198006892408570660640278146190727489071849348493245293815426717} a^{12} + \frac{4739180263517824123490617369400236601676942017093310323624125543278766989832259810712288726682612117891995115}{835009510396320595361055059946475043484240810518376082708902847927683337754288729868862192181918943525785120604} a^{11} + \frac{38340594391493672563910225596973031814997142427760170611288771413497963570077124642626691596168685273069707619}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{10} + \frac{265042789822138629398127854260473707694608853465464048178706701235845136190081342406618424810557096693439717574}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{9} - \frac{98439741135027959657856695642486407351294781872002330452912472034268356357268160352730094097578826585863838075}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{8} - \frac{300385633760254545310850222043543919077563936607541893738182039221556528648128684265336299996537370256602417572}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{7} + \frac{188673402733189699937356482844461665715946690985617146605203140844132248233189939366766531058008010454984486720}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{6} - \frac{8931100443515036024624253948799796738652854761360260936544541259348368739453932506339389073755804139549822059}{46389417244240033075614169997026391304680045028798671261605713773760185430793818326047899565662163529210284478} a^{5} + \frac{251984790898403272587547780522059214001880846579179684875677306510460835477337853516043248090199322583297830458}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a^{4} + \frac{295030056676565440342448096823357859928997986258738558474405451423542536422800744813981667625985684603478242111}{835009510396320595361055059946475043484240810518376082708902847927683337754288729868862192181918943525785120604} a^{3} - \frac{721959805642776172521925942896759431535189555583055661519499816126347120076385821040759185255245572575240699313}{2505028531188961786083165179839425130452722431555128248126708543783050013262866189606586576545756830577355361812} a^{2} + \frac{176328042408492276476039121039945321383757840045107319933677475033267245408984843398770159280198632346893881056}{626257132797240446520791294959856282613180607888782062031677135945762503315716547401646644136439207644338840453} a - \frac{323439722148390927857532416355815860344420585056267734640111179977820039870302613078642020475351663501439228981}{1252514265594480893041582589919712565226361215777564124063354271891525006631433094803293288272878415288677680906}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{5562}$, which has order $1601856$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 4097659676.87 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 80 |
| The 14 conjugacy class representatives for $C_4:F_5$ |
| Character table for $C_4:F_5$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.0.4166224.1, 5.5.2382032.1, 10.10.8056377164681869568.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ | R | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $17$ | 17.4.3.2 | $x^{4} - 153$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 17.8.6.1 | $x^{8} - 119 x^{4} + 23409$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 53 | Data not computed | ||||||