Normalized defining polynomial
\( x^{26} - x^{25} - 128 x^{24} + 120 x^{23} + 6463 x^{22} - 5975 x^{21} - 170783 x^{20} + 151479 x^{19} + 2653408 x^{18} - 2161208 x^{17} - 25843163 x^{16} + 18764532 x^{15} + 163067279 x^{14} - 104314451 x^{13} - 672763111 x^{12} + 382039090 x^{11} + 1790670843 x^{10} - 929930603 x^{9} - 2949081397 x^{8} + 1478044102 x^{7} + 2744646036 x^{6} - 1431320998 x^{5} - 1163844385 x^{4} + 695470618 x^{3} + 86015551 x^{2} - 84268697 x + 8312527 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[26, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(136256962349903875302212256879702783874125319329547338554073=3^{13}\cdot 131^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $188.10$ | 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: | $3, 131$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(393=3\cdot 131\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{393}(1,·)$, $\chi_{393}(322,·)$, $\chi_{393}(68,·)$, $\chi_{393}(325,·)$, $\chi_{393}(193,·)$, $\chi_{393}(392,·)$, $\chi_{393}(211,·)$, $\chi_{393}(149,·)$, $\chi_{393}(86,·)$, $\chi_{393}(281,·)$, $\chi_{393}(346,·)$, $\chi_{393}(155,·)$, $\chi_{393}(92,·)$, $\chi_{393}(32,·)$, $\chi_{393}(361,·)$, $\chi_{393}(71,·)$, $\chi_{393}(301,·)$, $\chi_{393}(238,·)$, $\chi_{393}(47,·)$, $\chi_{393}(112,·)$, $\chi_{393}(200,·)$, $\chi_{393}(307,·)$, $\chi_{393}(52,·)$, $\chi_{393}(182,·)$, $\chi_{393}(244,·)$, $\chi_{393}(341,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{61} a^{21} + \frac{15}{61} a^{19} + \frac{17}{61} a^{18} - \frac{29}{61} a^{17} - \frac{13}{61} a^{16} + \frac{10}{61} a^{15} - \frac{4}{61} a^{14} - \frac{23}{61} a^{13} - \frac{23}{61} a^{12} + \frac{27}{61} a^{11} - \frac{28}{61} a^{10} - \frac{7}{61} a^{9} + \frac{22}{61} a^{8} - \frac{29}{61} a^{7} + \frac{3}{61} a^{6} - \frac{15}{61} a^{5} + \frac{9}{61} a^{4} - \frac{15}{61} a^{2} - \frac{11}{61} a - \frac{29}{61}$, $\frac{1}{61} a^{22} + \frac{15}{61} a^{20} + \frac{17}{61} a^{19} - \frac{29}{61} a^{18} - \frac{13}{61} a^{17} + \frac{10}{61} a^{16} - \frac{4}{61} a^{15} - \frac{23}{61} a^{14} - \frac{23}{61} a^{13} + \frac{27}{61} a^{12} - \frac{28}{61} a^{11} - \frac{7}{61} a^{10} + \frac{22}{61} a^{9} - \frac{29}{61} a^{8} + \frac{3}{61} a^{7} - \frac{15}{61} a^{6} + \frac{9}{61} a^{5} - \frac{15}{61} a^{3} - \frac{11}{61} a^{2} - \frac{29}{61} a$, $\frac{1}{10553} a^{23} + \frac{68}{10553} a^{22} - \frac{31}{10553} a^{21} - \frac{69}{173} a^{20} - \frac{3833}{10553} a^{19} + \frac{5041}{10553} a^{18} - \frac{28}{10553} a^{17} - \frac{4094}{10553} a^{16} + \frac{465}{10553} a^{15} - \frac{19}{173} a^{14} - \frac{5237}{10553} a^{13} + \frac{5245}{10553} a^{12} - \frac{2238}{10553} a^{11} - \frac{2277}{10553} a^{10} - \frac{2542}{10553} a^{9} - \frac{5116}{10553} a^{8} + \frac{3841}{10553} a^{7} + \frac{498}{10553} a^{6} - \frac{2358}{10553} a^{5} + \frac{1767}{10553} a^{4} + \frac{1043}{10553} a^{3} + \frac{767}{10553} a^{2} - \frac{3418}{10553} a + \frac{1456}{10553}$, $\frac{1}{80541139733} a^{24} - \frac{2382431}{80541139733} a^{23} - \frac{25949096}{80541139733} a^{22} - \frac{527198564}{80541139733} a^{21} + \frac{5761424893}{80541139733} a^{20} - \frac{28659672489}{80541139733} a^{19} + \frac{22100990945}{80541139733} a^{18} + \frac{26673809702}{80541139733} a^{17} + \frac{20963337591}{80541139733} a^{16} - \frac{7609437782}{80541139733} a^{15} + \frac{24845826438}{80541139733} a^{14} + \frac{33226726836}{80541139733} a^{13} - \frac{14506481084}{80541139733} a^{12} - \frac{17100918094}{80541139733} a^{11} + \frac{30273953883}{80541139733} a^{10} - \frac{25568596434}{80541139733} a^{9} + \frac{8731972345}{80541139733} a^{8} + \frac{17799463527}{80541139733} a^{7} - \frac{28410557989}{80541139733} a^{6} + \frac{14998918229}{80541139733} a^{5} + \frac{27525562073}{80541139733} a^{4} + \frac{13988693860}{80541139733} a^{3} + \frac{39173652892}{80541139733} a^{2} + \frac{29856413742}{80541139733} a - \frac{524462178}{1320346553}$, $\frac{1}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{25} + \frac{856494438109526421580125317902189822291065281143160461968760456144931785}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{24} - \frac{20785382926254822038145258879531406010651708057540285717427591476719371965939}{11223582122227910877749558144281679347611271415276154178454063874688031056804680581} a^{23} + \frac{1854136638479782768339564215936168941813663355198740015348116853099903368084728855}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{22} + \frac{682850611776967392428030103288429938329959811272120797182418493809810473681419696}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{21} + \frac{76606092606790183025502210403288431608330285286663140651073098977677077771659009472}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{20} - \frac{169410689774255920268221250831260853887813703622759384857745379554694250477347130146}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{19} + \frac{6377735506066527862634163176138597349624433126714609620336973121268792121992667650}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{18} + \frac{124934467098712534970091744632771724007419283585292936057996561244148638012704332366}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{17} - \frac{125804246234787833566862046400615288398684132112190871659261971595530250286853385549}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{16} + \frac{299602765672203308787030926227676808233401282401778877194235734366229261179808413203}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{15} + \frac{91666187932154635694783699068488125591397342700856081043678767902356862217323730921}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{14} - \frac{284048471476817245969414111795895288794844653390325332383334205240786570254662274938}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{13} + \frac{324937955159411477521264426451745505810040017878971146301907760105975583751101061045}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{12} - \frac{200873230144628954832670476382774514956981782052273490377062478873776240662913523436}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{11} + \frac{291368090591343183319240353018701769759293675614326253166017943723910156907315016885}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{10} - \frac{57806743199846483421612169418719386346200134448562504372732523577293645298247446739}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{9} + \frac{205847304805817721899136653245381188850931159631009278626561781897791243131722216503}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{8} - \frac{37349347934417249846467838506041330292102263679546444747834349648991827437817670248}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{7} + \frac{125129956339438661628498019265160816946863549957232508884802406758177731557200034147}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{6} + \frac{198253901459437876587666688503516351481901900538374507208978501299057664470309615801}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{5} - \frac{341446090135172263207764561982322957756772345935648520581580828673242146275618983326}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{4} - \frac{226692257857078512421818832202949368811746710658755778151014034682062038288132317687}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{3} + \frac{30793245755534165642782754496516589996516428543283111594962913195675143185168961538}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a^{2} - \frac{84501282926712157187540598394126446756511981063614329271896098254498215252521369770}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441} a + \frac{48192098072927432329518527617936583476074044197104263531192532504682998754492254049}{684638509455902563542723046801182440204287556331845404885697896355969894465085515441}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $25$ | 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: | \( 1768130091528311500000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{393}) \), 13.13.25542038069936263923006961.1 |
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 | ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ | R | $26$ | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ | $26$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 131 | Data not computed | ||||||