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Properties

Label 26.26.1919641021...8128.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{39}\cdot 79^{24}$
Root discriminant $159.66$
Ramified primes $2, 79$
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![-35447, -50580, 3461310, -8099910, -33898995, 153068810, -147518415, -168890324, 375988637, -49085562, -299732504, 151824110, 103695959, -87417128, -13363361, 23968180, -1089717, -3564090, 533589, 293118, -64015, -12928, 3600, 274, -97, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)
 
gp: K = bnfinit(x^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447, 1)
 

Normalized defining polynomial

\( x^{26} - 2 x^{25} - 97 x^{24} + 274 x^{23} + 3600 x^{22} - 12928 x^{21} - 64015 x^{20} + 293118 x^{19} + 533589 x^{18} - 3564090 x^{17} - 1089717 x^{16} + 23968180 x^{15} - 13363361 x^{14} - 87417128 x^{13} + 103695959 x^{12} + 151824110 x^{11} - 299732504 x^{10} - 49085562 x^{9} + 375988637 x^{8} - 168890324 x^{7} - 147518415 x^{6} + 153068810 x^{5} - 33898995 x^{4} - 8099910 x^{3} + 3461310 x^{2} - 50580 x - 35447 \)

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:  \(1919641021107487828653877081110544587155912070949008048128=2^{39}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $159.66$
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:  $2, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(632=2^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(97,·)$, $\chi_{632}(65,·)$, $\chi_{632}(457,·)$, $\chi_{632}(141,·)$, $\chi_{632}(337,·)$, $\chi_{632}(405,·)$, $\chi_{632}(89,·)$, $\chi_{632}(413,·)$, $\chi_{632}(289,·)$, $\chi_{632}(101,·)$, $\chi_{632}(433,·)$, $\chi_{632}(617,·)$, $\chi_{632}(225,·)$, $\chi_{632}(125,·)$, $\chi_{632}(301,·)$, $\chi_{632}(541,·)$, $\chi_{632}(561,·)$, $\chi_{632}(117,·)$, $\chi_{632}(417,·)$, $\chi_{632}(381,·)$, $\chi_{632}(441,·)$, $\chi_{632}(21,·)$, $\chi_{632}(317,·)$, $\chi_{632}(605,·)$, $\chi_{632}(245,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{6}{23} a^{15} - \frac{11}{23} a^{14} + \frac{11}{23} a^{13} - \frac{2}{23} a^{11} + \frac{3}{23} a^{10} - \frac{7}{23} a^{9} + \frac{1}{23} a^{8} - \frac{10}{23} a^{7} + \frac{7}{23} a^{6} - \frac{11}{23} a^{5} + \frac{1}{23} a^{4} - \frac{8}{23} a^{3} - \frac{5}{23} a^{2} + \frac{9}{23} a + \frac{4}{23}$, $\frac{1}{23} a^{18} - \frac{6}{23} a^{16} - \frac{11}{23} a^{15} + \frac{11}{23} a^{14} - \frac{2}{23} a^{12} + \frac{3}{23} a^{11} - \frac{7}{23} a^{10} + \frac{1}{23} a^{9} - \frac{10}{23} a^{8} + \frac{7}{23} a^{7} - \frac{11}{23} a^{6} + \frac{1}{23} a^{5} - \frac{8}{23} a^{4} - \frac{5}{23} a^{3} + \frac{9}{23} a^{2} + \frac{4}{23} a$, $\frac{1}{23} a^{19} - \frac{11}{23} a^{16} - \frac{2}{23} a^{15} + \frac{3}{23} a^{14} - \frac{5}{23} a^{13} + \frac{3}{23} a^{12} + \frac{4}{23} a^{11} - \frac{4}{23} a^{10} - \frac{6}{23} a^{9} - \frac{10}{23} a^{8} - \frac{2}{23} a^{7} - \frac{3}{23} a^{6} - \frac{5}{23} a^{5} + \frac{1}{23} a^{4} + \frac{7}{23} a^{3} - \frac{3}{23} a^{2} + \frac{8}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{20} - \frac{2}{23} a^{16} + \frac{6}{23} a^{15} - \frac{11}{23} a^{14} + \frac{9}{23} a^{13} + \frac{4}{23} a^{12} - \frac{3}{23} a^{11} + \frac{4}{23} a^{10} + \frac{5}{23} a^{9} + \frac{9}{23} a^{8} + \frac{2}{23} a^{7} + \frac{3}{23} a^{6} - \frac{5}{23} a^{5} - \frac{5}{23} a^{4} + \frac{1}{23} a^{3} - \frac{1}{23} a^{2} + \frac{8}{23} a - \frac{2}{23}$, $\frac{1}{23} a^{21} + \frac{6}{23} a^{16} + \frac{10}{23} a^{14} + \frac{3}{23} a^{13} - \frac{3}{23} a^{12} + \frac{11}{23} a^{10} - \frac{5}{23} a^{9} + \frac{4}{23} a^{8} + \frac{6}{23} a^{7} + \frac{9}{23} a^{6} - \frac{4}{23} a^{5} + \frac{3}{23} a^{4} + \frac{6}{23} a^{3} - \frac{2}{23} a^{2} - \frac{7}{23} a + \frac{8}{23}$, $\frac{1}{2369} a^{22} - \frac{17}{2369} a^{21} + \frac{4}{2369} a^{20} + \frac{9}{2369} a^{19} + \frac{1}{103} a^{18} + \frac{33}{2369} a^{17} - \frac{1014}{2369} a^{16} - \frac{882}{2369} a^{15} + \frac{830}{2369} a^{14} - \frac{893}{2369} a^{13} + \frac{623}{2369} a^{12} + \frac{326}{2369} a^{11} - \frac{453}{2369} a^{10} - \frac{1031}{2369} a^{9} - \frac{664}{2369} a^{8} - \frac{1017}{2369} a^{7} + \frac{1144}{2369} a^{6} + \frac{951}{2369} a^{5} + \frac{293}{2369} a^{4} - \frac{48}{103} a^{3} + \frac{436}{2369} a^{2} - \frac{745}{2369} a + \frac{1054}{2369}$, $\frac{1}{2369} a^{23} + \frac{24}{2369} a^{21} - \frac{26}{2369} a^{20} - \frac{30}{2369} a^{19} + \frac{12}{2369} a^{18} - \frac{41}{2369} a^{17} + \frac{523}{2369} a^{16} - \frac{465}{2369} a^{15} + \frac{651}{2369} a^{14} + \frac{480}{2369} a^{13} + \frac{308}{2369} a^{12} + \frac{145}{2369} a^{11} - \frac{801}{2369} a^{10} - \frac{990}{2369} a^{9} - \frac{666}{2369} a^{8} + \frac{232}{2369} a^{7} + \frac{108}{2369} a^{6} - \frac{20}{2369} a^{5} - \frac{655}{2369} a^{4} - \frac{307}{2369} a^{3} + \frac{1002}{2369} a^{2} + \frac{28}{2369} a + \frac{717}{2369}$, $\frac{1}{1253201} a^{24} + \frac{185}{1253201} a^{23} - \frac{31}{1253201} a^{22} + \frac{7512}{1253201} a^{21} - \frac{26072}{1253201} a^{20} - \frac{12522}{1253201} a^{19} - \frac{16081}{1253201} a^{18} - \frac{26902}{1253201} a^{17} - \frac{188149}{1253201} a^{16} + \frac{457845}{1253201} a^{15} - \frac{618234}{1253201} a^{14} + \frac{211456}{1253201} a^{13} - \frac{60261}{1253201} a^{12} - \frac{370122}{1253201} a^{11} + \frac{304323}{1253201} a^{10} + \frac{35423}{1253201} a^{9} - \frac{296475}{1253201} a^{8} - \frac{180373}{1253201} a^{7} + \frac{147693}{1253201} a^{6} + \frac{615930}{1253201} a^{5} + \frac{300977}{1253201} a^{4} + \frac{213811}{1253201} a^{3} - \frac{478830}{1253201} a^{2} + \frac{623878}{1253201} a - \frac{5925}{12167}$, $\frac{1}{449646000489879865723294746707522513589060530249361489389929} a^{25} + \frac{7424153701767750673929186247585716954675018729772581}{19549826108255646335795423769892283199524370880407021277823} a^{24} - \frac{72058753853644062064028255503710003928025138453555243432}{449646000489879865723294746707522513589060530249361489389929} a^{23} + \frac{86089664136669273438726585409803418403811197509075471095}{449646000489879865723294746707522513589060530249361489389929} a^{22} + \frac{8995068199417140480256485345890236757160627772293193953520}{449646000489879865723294746707522513589060530249361489389929} a^{21} - \frac{4383008153042660810791923243578241013425878194479838301789}{449646000489879865723294746707522513589060530249361489389929} a^{20} - \frac{5355696140379732845828931824479583095105413418899782247803}{449646000489879865723294746707522513589060530249361489389929} a^{19} + \frac{1346878813650853971593251995243834338401248031162755703479}{449646000489879865723294746707522513589060530249361489389929} a^{18} + \frac{3466719936378292617540349521786967547313094861775424915156}{449646000489879865723294746707522513589060530249361489389929} a^{17} - \frac{206380992384711654254475762710917887195349622569979120182695}{449646000489879865723294746707522513589060530249361489389929} a^{16} - \frac{113717465506317794764468113203586508102514447722860535014373}{449646000489879865723294746707522513589060530249361489389929} a^{15} - \frac{84352060853200902085877364636968569760071996595541967120109}{449646000489879865723294746707522513589060530249361489389929} a^{14} - \frac{100967880039464311290746437910310838932704906610500709324013}{449646000489879865723294746707522513589060530249361489389929} a^{13} - \frac{113937522438909384518599086972476683872153848811999840503196}{449646000489879865723294746707522513589060530249361489389929} a^{12} - \frac{218880179804883816134204699491400610907441506986393046568615}{449646000489879865723294746707522513589060530249361489389929} a^{11} + \frac{151043081843683391642318955001617573285154540770691693897323}{449646000489879865723294746707522513589060530249361489389929} a^{10} - \frac{150457099043214347343256696760796513927930339817258536195219}{449646000489879865723294746707522513589060530249361489389929} a^{9} + \frac{159747008313894924837453822747020977836207974537976442041153}{449646000489879865723294746707522513589060530249361489389929} a^{8} - \frac{9921280530747583085580135873389965538210175107885652319771}{449646000489879865723294746707522513589060530249361489389929} a^{7} + \frac{179276962794917060463431164542874483668045524589403779486529}{449646000489879865723294746707522513589060530249361489389929} a^{6} + \frac{208498242753106435837244415305462567858196004624775395241528}{449646000489879865723294746707522513589060530249361489389929} a^{5} - \frac{28957418614289990932796743054609214072840354181022831715052}{449646000489879865723294746707522513589060530249361489389929} a^{4} + \frac{198646376408222605383817644029203530290788131532583951929598}{449646000489879865723294746707522513589060530249361489389929} a^{3} + \frac{31525673914515712056929579181936080817721406550270814577321}{449646000489879865723294746707522513589060530249361489389929} a^{2} - \frac{30747074909120844702041806448395251326327214984837710910368}{449646000489879865723294746707522513589060530249361489389929} a + \frac{221458518519118340765955661450975364003416941584579811472312}{449646000489879865723294746707522513589060530249361489389929}$

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:  \( 514750610132899270000 \) (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

\(\Q(\sqrt{2}) \), 13.13.59091511031674153381441.1

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 R $26$ $26$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ $26$ $26$

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
2Data not computed
$79$79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$