Properties

Label 26.0.463...123.1
Degree $26$
Signature $[0, 13]$
Discriminant $-4.635\times 10^{42}$
Root discriminant $43.75$
Ramified primes $3, 1093$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171)
 
gp: K = bnfinit(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2259171, 4678236, 1301535, -4130865, -4211379, -188640, 1978515, 873384, -55703, 189642, 37881, -318218, -161565, 64299, 59660, 13671, 1038, -5528, -4404, -27, 794, 186, -12, -20, -9, 0, 1]);
 

\( x^{26} - 9 x^{24} - 20 x^{23} - 12 x^{22} + 186 x^{21} + 794 x^{20} - 27 x^{19} - 4404 x^{18} - 5528 x^{17} + 1038 x^{16} + 13671 x^{15} + 59660 x^{14} + 64299 x^{13} - 161565 x^{12} - 318218 x^{11} + 37881 x^{10} + 189642 x^{9} - 55703 x^{8} + 873384 x^{7} + 1978515 x^{6} - 188640 x^{5} - 4211379 x^{4} - 4130865 x^{3} + 1301535 x^{2} + 4678236 x + 2259171 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-4634664002656599036916177705626061158525123\)\(\medspace = -\,3^{13}\cdot 1093^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.75$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 1093$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{8}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} + \frac{8}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{1}{27} a^{6}$, $\frac{1}{81} a^{17} + \frac{1}{27} a^{13} + \frac{1}{81} a^{11} + \frac{1}{27} a^{9} - \frac{4}{27} a^{7} + \frac{25}{81} a^{5} + \frac{8}{27} a^{3} + \frac{4}{9} a$, $\frac{1}{891} a^{18} + \frac{2}{891} a^{17} - \frac{4}{297} a^{16} - \frac{5}{297} a^{15} - \frac{4}{99} a^{14} - \frac{1}{33} a^{13} + \frac{34}{891} a^{12} - \frac{13}{891} a^{11} - \frac{10}{297} a^{10} + \frac{7}{297} a^{9} - \frac{2}{33} a^{8} - \frac{8}{99} a^{7} + \frac{127}{891} a^{6} - \frac{151}{891} a^{5} - \frac{148}{297} a^{4} - \frac{10}{27} a^{3} + \frac{28}{99} a^{2} - \frac{34}{99} a + \frac{2}{11}$, $\frac{1}{891} a^{19} - \frac{5}{891} a^{17} + \frac{1}{99} a^{16} - \frac{2}{297} a^{15} + \frac{5}{99} a^{14} + \frac{2}{81} a^{13} + \frac{2}{99} a^{12} + \frac{7}{891} a^{11} - \frac{2}{99} a^{10} + \frac{4}{99} a^{9} + \frac{4}{99} a^{8} - \frac{59}{891} a^{7} + \frac{10}{99} a^{6} + \frac{430}{891} a^{5} + \frac{40}{99} a^{4} - \frac{136}{297} a^{3} - \frac{8}{33} a^{2} - \frac{2}{99} a - \frac{4}{11}$, $\frac{1}{2673} a^{20} - \frac{1}{2673} a^{19} - \frac{1}{297} a^{17} + \frac{8}{891} a^{16} + \frac{1}{297} a^{15} + \frac{94}{2673} a^{14} - \frac{7}{2673} a^{13} - \frac{13}{891} a^{12} + \frac{4}{99} a^{11} - \frac{32}{891} a^{10} + \frac{8}{297} a^{9} - \frac{68}{2673} a^{8} - \frac{46}{2673} a^{7} + \frac{94}{891} a^{6} + \frac{8}{27} a^{5} + \frac{53}{297} a^{4} - \frac{7}{33} a^{3} - \frac{34}{99} a^{2} - \frac{13}{99} a + \frac{1}{11}$, $\frac{1}{8019} a^{21} + \frac{1}{8019} a^{20} - \frac{2}{8019} a^{19} + \frac{8}{2673} a^{17} + \frac{16}{2673} a^{16} + \frac{76}{8019} a^{15} - \frac{31}{729} a^{14} + \frac{100}{8019} a^{13} + \frac{145}{2673} a^{12} - \frac{65}{2673} a^{11} - \frac{64}{2673} a^{10} + \frac{166}{8019} a^{9} + \frac{421}{8019} a^{8} - \frac{854}{8019} a^{7} - \frac{388}{2673} a^{6} - \frac{98}{891} a^{5} + \frac{61}{891} a^{4} + \frac{5}{33} a^{3} + \frac{5}{11} a^{2} + \frac{112}{297} a + \frac{8}{33}$, $\frac{1}{8019} a^{22} - \frac{1}{8019} a^{19} - \frac{1}{2673} a^{18} + \frac{14}{2673} a^{17} + \frac{127}{8019} a^{16} + \frac{5}{2673} a^{15} - \frac{128}{2673} a^{14} - \frac{442}{8019} a^{13} - \frac{20}{891} a^{12} - \frac{38}{2673} a^{11} + \frac{26}{729} a^{10} + \frac{67}{2673} a^{9} - \frac{205}{2673} a^{8} - \frac{53}{729} a^{7} - \frac{371}{2673} a^{6} + \frac{62}{891} a^{5} + \frac{179}{891} a^{4} + \frac{49}{297} a^{3} + \frac{118}{297} a^{2} + \frac{62}{297} a + \frac{10}{33}$, $\frac{1}{24057} a^{23} - \frac{1}{24057} a^{22} - \frac{1}{24057} a^{20} + \frac{7}{24057} a^{19} + \frac{1}{2673} a^{18} - \frac{131}{24057} a^{17} + \frac{104}{24057} a^{16} + \frac{29}{8019} a^{15} - \frac{1327}{24057} a^{14} + \frac{1135}{24057} a^{13} - \frac{134}{8019} a^{12} - \frac{950}{24057} a^{11} + \frac{239}{24057} a^{10} + \frac{376}{8019} a^{9} - \frac{2749}{24057} a^{8} + \frac{46}{24057} a^{7} - \frac{4}{8019} a^{6} + \frac{259}{891} a^{5} - \frac{173}{2673} a^{4} + \frac{128}{297} a^{3} + \frac{229}{891} a^{2} + \frac{298}{891} a - \frac{13}{99}$, $\frac{1}{697653} a^{24} + \frac{1}{63423} a^{23} - \frac{13}{232551} a^{22} - \frac{2}{63423} a^{21} - \frac{107}{697653} a^{20} + \frac{1}{2871} a^{19} - \frac{212}{697653} a^{18} - \frac{1729}{697653} a^{17} - \frac{8}{232551} a^{16} + \frac{343}{24057} a^{15} + \frac{25339}{697653} a^{14} + \frac{8224}{232551} a^{13} + \frac{12442}{697653} a^{12} - \frac{15166}{697653} a^{11} - \frac{2855}{77517} a^{10} + \frac{11756}{697653} a^{9} + \frac{110740}{697653} a^{8} + \frac{1604}{232551} a^{7} + \frac{197}{2871} a^{6} - \frac{26393}{77517} a^{5} + \frac{1904}{8613} a^{4} + \frac{1399}{25839} a^{3} + \frac{2011}{25839} a^{2} - \frac{140}{957} a - \frac{84}{319}$, $\frac{1}{81532389886724403571936116880489530454479711} a^{25} - \frac{56139696771393469470078943515625058918}{81532389886724403571936116880489530454479711} a^{24} - \frac{1022436181847995615356088599410464433897}{81532389886724403571936116880489530454479711} a^{23} + \frac{1499627502792998190703703851200014622542}{81532389886724403571936116880489530454479711} a^{22} - \frac{4783584305954396314330410923462447008210}{81532389886724403571936116880489530454479711} a^{21} - \frac{1240232907110849259068322884638153436821}{81532389886724403571936116880489530454479711} a^{20} - \frac{8130054289359684157309011202213884334601}{81532389886724403571936116880489530454479711} a^{19} - \frac{31617025266451232047074267891935731756733}{81532389886724403571936116880489530454479711} a^{18} - \frac{231386328170740317532053382872664341451868}{81532389886724403571936116880489530454479711} a^{17} - \frac{339513753044780508099511159172908401721141}{81532389886724403571936116880489530454479711} a^{16} + \frac{964000215617017679958006920992752586473728}{81532389886724403571936116880489530454479711} a^{15} - \frac{3299818074689503809870289742284669170234304}{81532389886724403571936116880489530454479711} a^{14} - \frac{3270330785687954206341135904963896998256830}{81532389886724403571936116880489530454479711} a^{13} - \frac{195815788799725034392849305707690004073180}{7412035444247673051994192443680866404952701} a^{12} + \frac{2963533154359106565630282972025862912052952}{81532389886724403571936116880489530454479711} a^{11} - \frac{2089731924310030740353049719468331000094432}{81532389886724403571936116880489530454479711} a^{10} + \frac{22990411929013046719003884575098325818028}{1896102090388939617952002718150919312894877} a^{9} + \frac{10270045632509183114589945045557661166138604}{81532389886724403571936116880489530454479711} a^{8} + \frac{2287865854310620760587301594238991835476171}{27177463295574801190645372293496510151493237} a^{7} - \frac{640651972033487850962725561392289068767420}{9059154431858267063548457431165503383831079} a^{6} + \frac{1051186582342645786362314111254440132565424}{9059154431858267063548457431165503383831079} a^{5} - \frac{326945052818760509208787248355166575642307}{3019718143952755687849485810388501127943693} a^{4} + \frac{227301349998375404394209981847805877169974}{1006572714650918562616495270129500375981231} a^{3} - \frac{56875137735393937697153054638850113947346}{3019718143952755687849485810388501127943693} a^{2} + \frac{162213268691348358943579540551237746182118}{335524238216972854205498423376500125327077} a - \frac{8555948587850113940220145211874441253741}{37280470912996983800610935930722236147453}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{36208078418198207877821}{4458879115785042788098908891} a^{25} - \frac{359615535410293894115}{45039182987727704930292009} a^{24} - \frac{31262516208027746085383}{495431012865004754233212099} a^{23} - \frac{425990883981092931175735}{4458879115785042788098908891} a^{22} - \frac{22199755028563812957317}{1486293038595014262699636297} a^{21} + \frac{2169545031902070785718007}{1486293038595014262699636297} a^{20} + \frac{1966809765563692650586139}{405352646889549344372628081} a^{19} - \frac{2436372673414889562011809}{495431012865004754233212099} a^{18} - \frac{43113806312603747638121330}{1486293038595014262699636297} a^{17} - \frac{55496681384568909869166544}{4458879115785042788098908891} a^{16} + \frac{27681912439440856669971023}{1486293038595014262699636297} a^{15} + \frac{12712401746592951227598830}{165143670955001584744404033} a^{14} + \frac{1692552223897118071205295928}{4458879115785042788098908891} a^{13} + \frac{168498381437229074001685235}{1486293038595014262699636297} a^{12} - \frac{2028905847961002529244927770}{1486293038595014262699636297} a^{11} - \frac{4313244339113148598844038264}{4458879115785042788098908891} a^{10} + \frac{720907275691409161955384764}{495431012865004754233212099} a^{9} - \frac{19088612661357788993348372}{135117548963183114790876027} a^{8} - \frac{260934342760137406650214223}{405352646889549344372628081} a^{7} + \frac{10759599472953414157159779574}{1486293038595014262699636297} a^{6} + \frac{336473067120924159363559183}{45039182987727704930292009} a^{5} - \frac{4829827678413257911798921813}{495431012865004754233212099} a^{4} - \frac{3635339375171236108272860357}{165143670955001584744404033} a^{3} - \frac{1174528933481695845663433988}{165143670955001584744404033} a^{2} + \frac{3149028082692286688645040485}{165143670955001584744404033} a + \frac{296381255429909873085936859}{18349296772777953860489337} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 16355807538960.088 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 16355807538960.088 \cdot 1}{6\sqrt{4634664002656599036916177705626061158525123}}\approx 30.1196487161707$ (assuming GRH)

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 13.1.1242935235998051916321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling: Deg 26

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ R $26$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ $26$ $26$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
1093Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1093.2t1.a.a$1$ $ 1093 $ \(\Q(\sqrt{1093}) \) $C_2$ (as 2T1) $1$ $1$
1.3279.2t1.a.a$1$ $ 3 \cdot 1093 $ \(\Q(\sqrt{-3279}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3279.26t3.a.b$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3279.13t2.a.a$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.13t2.a.f$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.26t3.a.d$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3279.26t3.a.c$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3279.13t2.a.c$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.13t2.a.e$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.13t2.a.b$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.26t3.a.f$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3279.26t3.a.e$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3279.13t2.a.d$2$ $ 3 \cdot 1093 $ 13.1.1242935235998051916321.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3279.26t3.a.a$2$ $ 3 \cdot 1093 $ 26.0.4634664002656599036916177705626061158525123.1 $D_{26}$ (as 26T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.