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Properties

Label 26.26.7963814804...3125.1
Degree $26$
Signature $[26, 0]$
Discriminant $5^{13}\cdot 131^{24}$
Root discriminant $201.32$
Ramified primes $5, 131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5379961, -153269122, 777425706, -300524389, -4634451406, 8050593245, 1563567289, -12721612999, 5497185211, 7250022480, -5626894204, -1595204916, 2264921609, -14946243, -469879785, 69443954, 53470266, -13246818, -3213380, 1172291, 78438, -54186, 924, 1244, -79, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 - 79*x^24 + 1244*x^23 + 924*x^22 - 54186*x^21 + 78438*x^20 + 1172291*x^19 - 3213380*x^18 - 13246818*x^17 + 53470266*x^16 + 69443954*x^15 - 469879785*x^14 - 14946243*x^13 + 2264921609*x^12 - 1595204916*x^11 - 5626894204*x^10 + 7250022480*x^9 + 5497185211*x^8 - 12721612999*x^7 + 1563567289*x^6 + 8050593245*x^5 - 4634451406*x^4 - 300524389*x^3 + 777425706*x^2 - 153269122*x + 5379961)
 
gp: K = bnfinit(x^26 - 11*x^25 - 79*x^24 + 1244*x^23 + 924*x^22 - 54186*x^21 + 78438*x^20 + 1172291*x^19 - 3213380*x^18 - 13246818*x^17 + 53470266*x^16 + 69443954*x^15 - 469879785*x^14 - 14946243*x^13 + 2264921609*x^12 - 1595204916*x^11 - 5626894204*x^10 + 7250022480*x^9 + 5497185211*x^8 - 12721612999*x^7 + 1563567289*x^6 + 8050593245*x^5 - 4634451406*x^4 - 300524389*x^3 + 777425706*x^2 - 153269122*x + 5379961, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} - 79 x^{24} + 1244 x^{23} + 924 x^{22} - 54186 x^{21} + 78438 x^{20} + 1172291 x^{19} - 3213380 x^{18} - 13246818 x^{17} + 53470266 x^{16} + 69443954 x^{15} - 469879785 x^{14} - 14946243 x^{13} + 2264921609 x^{12} - 1595204916 x^{11} - 5626894204 x^{10} + 7250022480 x^{9} + 5497185211 x^{8} - 12721612999 x^{7} + 1563567289 x^{6} + 8050593245 x^{5} - 4634451406 x^{4} - 300524389 x^{3} + 777425706 x^{2} - 153269122 x + 5379961 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(796381480427335725451692817091700442621259832099024658203125=5^{13}\cdot 131^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.32$
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:  $5, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(655=5\cdot 131\)
Dirichlet character group:    $\lbrace$$\chi_{655}(576,·)$, $\chi_{655}(1,·)$, $\chi_{655}(194,·)$, $\chi_{655}(324,·)$, $\chi_{655}(584,·)$, $\chi_{655}(394,·)$, $\chi_{655}(369,·)$, $\chi_{655}(211,·)$, $\chi_{655}(84,·)$, $\chi_{655}(346,·)$, $\chi_{655}(604,·)$, $\chi_{655}(506,·)$, $\chi_{655}(99,·)$, $\chi_{655}(39,·)$, $\chi_{655}(361,·)$, $\chi_{655}(301,·)$, $\chi_{655}(176,·)$, $\chi_{655}(456,·)$, $\chi_{655}(244,·)$, $\chi_{655}(374,·)$, $\chi_{655}(631,·)$, $\chi_{655}(569,·)$, $\chi_{655}(314,·)$, $\chi_{655}(636,·)$, $\chi_{655}(586,·)$, $\chi_{655}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{344341} a^{22} - \frac{51298}{344341} a^{21} + \frac{83785}{344341} a^{20} - \frac{39808}{344341} a^{19} + \frac{62909}{344341} a^{18} - \frac{130051}{344341} a^{17} + \frac{113258}{344341} a^{16} - \frac{11485}{344341} a^{15} - \frac{168811}{344341} a^{14} - \frac{85405}{344341} a^{13} + \frac{20795}{344341} a^{12} + \frac{140955}{344341} a^{11} + \frac{90416}{344341} a^{10} - \frac{131050}{344341} a^{9} + \frac{127255}{344341} a^{8} + \frac{117906}{344341} a^{7} - \frac{87267}{344341} a^{6} - \frac{123684}{344341} a^{5} - \frac{84666}{344341} a^{4} - \frac{28609}{344341} a^{3} + \frac{167503}{344341} a^{2} - \frac{99741}{344341} a - \frac{1445}{3869}$, $\frac{1}{344341} a^{23} + \frac{52903}{344341} a^{21} - \frac{101240}{344341} a^{20} - \frac{65745}{344341} a^{19} + \frac{156320}{344341} a^{18} + \frac{19594}{344341} a^{17} - \frac{168294}{344341} a^{16} - \frac{158890}{344341} a^{15} + \frac{79726}{344341} a^{14} - \frac{34352}{344341} a^{13} + \frac{114447}{344341} a^{12} - \frac{16653}{344341} a^{11} + \frac{99989}{344341} a^{10} + \frac{93698}{344341} a^{9} + \frac{28218}{344341} a^{8} - \frac{94944}{344341} a^{7} + \frac{31091}{344341} a^{6} + \frac{768}{344341} a^{5} - \frac{52044}{344341} a^{4} + \frac{164363}{344341} a^{3} + \frac{128180}{344341} a^{2} - \frac{79504}{344341} a + \frac{561}{3869}$, $\frac{1}{29655505980893777} a^{24} + \frac{406405297}{486155835752357} a^{23} + \frac{29048397715}{29655505980893777} a^{22} + \frac{13158126926776205}{29655505980893777} a^{21} + \frac{8681049828068107}{29655505980893777} a^{20} + \frac{5765003569599226}{29655505980893777} a^{19} + \frac{3404892881175427}{29655505980893777} a^{18} - \frac{12820571001439345}{29655505980893777} a^{17} - \frac{12890558277832427}{29655505980893777} a^{16} - \frac{11761520554465864}{29655505980893777} a^{15} + \frac{7708783773118132}{29655505980893777} a^{14} + \frac{11016000368580218}{29655505980893777} a^{13} + \frac{7555093278916077}{29655505980893777} a^{12} - \frac{1794705095737743}{29655505980893777} a^{11} - \frac{321740345244833}{29655505980893777} a^{10} + \frac{167221728788845}{406239807957449} a^{9} - \frac{10836125532409429}{29655505980893777} a^{8} + \frac{1002226300755145}{29655505980893777} a^{7} - \frac{3430610132676554}{29655505980893777} a^{6} + \frac{10617488315622406}{29655505980893777} a^{5} + \frac{7582396612711747}{29655505980893777} a^{4} - \frac{10162653685909813}{29655505980893777} a^{3} + \frac{1094824257193678}{29655505980893777} a^{2} - \frac{5413888630011947}{29655505980893777} a - \frac{99595154797841}{333207932369593}$, $\frac{1}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{25} - \frac{1203020843786774580398471989546820770200397909662260376475601630535}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{24} + \frac{107479794933136284818489516507480048226848991425916586390118116334044518116188}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{23} - \frac{30617506325648762401599791504718511612848993486509186752849547055313823420665}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{22} + \frac{40134273471020264217815184675557108934525490698490299194253328813772296194473929360}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{21} - \frac{23600347621492064177490879091744781450419197224801962229022749867739328565523561584}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{20} + \frac{42954519803616648776960727443486133400915139644691607075368823809051196568013766265}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{19} - \frac{30685695103200583211697607693488658307346038861531988750970533428953584320942325770}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{18} + \frac{78841535328643150914207105505893656968528475699627304894890022447171912372888990}{1425896518677932134555422811659785385989590430611604054139532914097085490657757153} a^{17} - \frac{42842439076356255568796733429831666235637572056038856210946294611147631396300113937}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{16} + \frac{26713998562398269201268811757405169100698462797565722218203665851173664886514292002}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{15} + \frac{23807770168088816327113496854193676465957448788392599851764981996450040533892986006}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{14} + \frac{4970492645623678204901675538759929780962731372985093064876706020522538491128507994}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{13} + \frac{4297199271986125350133317704686153111709498023925809925542828605585857388413852983}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{12} + \frac{25143740521717094034099263323694956695484597057628053884172149922506666150456808188}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{11} - \frac{11643762095013667866597628662140040495902959354329334523708944060558247854475680772}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{10} - \frac{3145329355417672111519337409394279152720607286288223655661867992503727085406361970}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{9} - \frac{25055326786441323669640480200065939924030494196933107754440188169543383228636594270}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{8} - \frac{8951951311046395331837042901113039059430960070808490127810546698382937604965460492}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{7} + \frac{3641881301076391161700282529169742290660875789623944266056325071305605457699702453}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{6} - \frac{3140539346040814174594210398642148873865697686177273174301510221704864871285057146}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{5} - \frac{3109648773219527151890946913090943984767384386827294104241952825095436230814622352}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{4} - \frac{8271125114171961823982047657092618132135718415532646826798815729619763245671714842}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{3} - \frac{869182481211478116371942025896276667581864931325483699570381926915142033791095496}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a^{2} + \frac{32005305751380397798934765631668409499544411874996129241336919764985587298437540871}{86979687639353860207880791511246908545365016267307847302511507759922214930123186333} a + \frac{43149504976224011021592369931330013188690915162155274722659484182209939255881192}{977299861116335507953716758553336051071517036711324126994511323145193426181159397}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 4932378447748938000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{26}$ (as 26T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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 $26$ $26$ R $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$131$131.13.12.1$x^{13} - 131$$13$$1$$12$$C_{13}$$[\ ]_{13}$
131.13.12.1$x^{13} - 131$$13$$1$$12$$C_{13}$$[\ ]_{13}$