Properties

Label 26.2.331...693.1
Degree $26$
Signature $[2, 12]$
Discriminant $3.317\times 10^{43}$
Root discriminant $47.19$
Ramified primes $13, 263$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 45*x^24 + 44*x^23 + 912*x^22 - 836*x^21 - 11055*x^20 + 9149*x^19 + 89968*x^18 - 63863*x^17 - 520434*x^16 + 297998*x^15 + 2198217*x^14 - 952653*x^13 - 6808591*x^12 + 2140242*x^11 + 15236220*x^10 - 3364016*x^9 - 23680109*x^8 + 2956997*x^7 + 23956149*x^6 - 618594*x^5 - 14015975*x^4 - 2419867*x^3 + 3438259*x^2 + 3561247*x - 1990843)
 
gp: K = bnfinit(x^26 - x^25 - 45*x^24 + 44*x^23 + 912*x^22 - 836*x^21 - 11055*x^20 + 9149*x^19 + 89968*x^18 - 63863*x^17 - 520434*x^16 + 297998*x^15 + 2198217*x^14 - 952653*x^13 - 6808591*x^12 + 2140242*x^11 + 15236220*x^10 - 3364016*x^9 - 23680109*x^8 + 2956997*x^7 + 23956149*x^6 - 618594*x^5 - 14015975*x^4 - 2419867*x^3 + 3438259*x^2 + 3561247*x - 1990843, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1990843, 3561247, 3438259, -2419867, -14015975, -618594, 23956149, 2956997, -23680109, -3364016, 15236220, 2140242, -6808591, -952653, 2198217, 297998, -520434, -63863, 89968, 9149, -11055, -836, 912, 44, -45, -1, 1]);
 

\( x^{26} - x^{25} - 45 x^{24} + 44 x^{23} + 912 x^{22} - 836 x^{21} - 11055 x^{20} + 9149 x^{19} + 89968 x^{18} - 63863 x^{17} - 520434 x^{16} + 297998 x^{15} + 2198217 x^{14} - 952653 x^{13} - 6808591 x^{12} + 2140242 x^{11} + 15236220 x^{10} - 3364016 x^{9} - 23680109 x^{8} + 2956997 x^{7} + 23956149 x^{6} - 618594 x^{5} - 14015975 x^{4} - 2419867 x^{3} + 3438259 x^{2} + 3561247 x - 1990843 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(33169014017307639762747533311185137385697693\)\(\medspace = 13^{13}\cdot 263^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $47.19$
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:  $13, 263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5} a^{19} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{35} a^{20} - \frac{2}{35} a^{19} - \frac{13}{35} a^{18} + \frac{12}{35} a^{17} + \frac{16}{35} a^{16} + \frac{2}{35} a^{15} - \frac{1}{35} a^{14} + \frac{1}{35} a^{13} + \frac{2}{35} a^{12} + \frac{3}{35} a^{11} - \frac{1}{35} a^{10} - \frac{17}{35} a^{9} - \frac{2}{35} a^{8} + \frac{2}{35} a^{6} - \frac{9}{35} a^{5} - \frac{17}{35} a^{4} + \frac{16}{35} a^{3} - \frac{3}{35} a - \frac{13}{35}$, $\frac{1}{35} a^{21} - \frac{3}{35} a^{19} - \frac{2}{5} a^{18} - \frac{2}{35} a^{17} + \frac{13}{35} a^{16} + \frac{2}{7} a^{15} + \frac{6}{35} a^{14} + \frac{4}{35} a^{13} - \frac{2}{5} a^{12} - \frac{9}{35} a^{11} - \frac{1}{7} a^{10} + \frac{13}{35} a^{9} - \frac{4}{35} a^{8} + \frac{9}{35} a^{7} + \frac{9}{35} a^{6} - \frac{2}{5} a^{5} + \frac{3}{35} a^{4} + \frac{11}{35} a^{3} + \frac{4}{35} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{175} a^{22} + \frac{1}{175} a^{21} + \frac{1}{175} a^{20} + \frac{2}{35} a^{19} - \frac{33}{175} a^{18} - \frac{46}{175} a^{17} - \frac{53}{175} a^{16} - \frac{81}{175} a^{15} - \frac{64}{175} a^{14} - \frac{6}{175} a^{13} + \frac{11}{35} a^{12} - \frac{37}{175} a^{11} + \frac{74}{175} a^{10} - \frac{59}{175} a^{9} - \frac{38}{175} a^{8} - \frac{52}{175} a^{7} - \frac{32}{175} a^{6} + \frac{58}{175} a^{5} + \frac{86}{175} a^{4} - \frac{61}{175} a^{3} - \frac{1}{175} a^{2} + \frac{83}{175} a + \frac{83}{175}$, $\frac{1}{3325} a^{23} + \frac{4}{3325} a^{22} + \frac{24}{3325} a^{21} + \frac{8}{3325} a^{20} + \frac{17}{3325} a^{19} - \frac{2}{665} a^{18} + \frac{1599}{3325} a^{17} + \frac{177}{665} a^{16} - \frac{607}{3325} a^{15} + \frac{662}{3325} a^{14} + \frac{116}{475} a^{13} - \frac{617}{3325} a^{12} + \frac{398}{3325} a^{11} + \frac{1188}{3325} a^{10} - \frac{13}{133} a^{9} - \frac{1461}{3325} a^{8} + \frac{27}{3325} a^{7} + \frac{377}{3325} a^{6} + \frac{96}{665} a^{5} + \frac{1497}{3325} a^{4} + \frac{26}{3325} a^{3} + \frac{214}{665} a^{2} - \frac{383}{3325} a + \frac{144}{3325}$, $\frac{1}{9679075} a^{24} - \frac{118}{1935815} a^{23} + \frac{15394}{9679075} a^{22} + \frac{12333}{9679075} a^{21} + \frac{91291}{9679075} a^{20} - \frac{27287}{509425} a^{19} - \frac{193234}{9679075} a^{18} + \frac{1459348}{9679075} a^{17} - \frac{504818}{1935815} a^{16} + \frac{1683734}{9679075} a^{15} + \frac{140209}{387163} a^{14} + \frac{2167574}{9679075} a^{13} - \frac{1730419}{9679075} a^{12} + \frac{4035539}{9679075} a^{11} + \frac{4035187}{9679075} a^{10} - \frac{685719}{1935815} a^{9} + \frac{4822863}{9679075} a^{8} + \frac{3172292}{9679075} a^{7} - \frac{410434}{1935815} a^{6} + \frac{23415}{55309} a^{5} + \frac{134699}{9679075} a^{4} + \frac{684834}{1935815} a^{3} + \frac{932411}{9679075} a^{2} - \frac{3859216}{9679075} a - \frac{4049468}{9679075}$, $\frac{1}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{25} - \frac{307734204975035330222659251167593657042233738616740530752991964751}{6161150727067134928401881415157759652111494801704180449552991084828521375} a^{24} + \frac{9992756646826747310647570835708028598964213376297531163798778703383546}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{23} - \frac{236247647911777292033197862923787215993219823261548633568203304794184149}{117061863814275563639635746887997433390118401232379428541506830611741906125} a^{22} + \frac{7415778875938656337710747968462540743137758142952853824942188719389599733}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{21} + \frac{481479527901747967957352773485870934754721475542193026412146418223787447}{43128055089469944498813169906104317564780463611929263146870937593799649625} a^{20} + \frac{4664883492098946759230052906581473259600991032055809470320768997725155549}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{19} + \frac{294806524558570959280823172520539978362039537234781524257657549094612887171}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{18} - \frac{311543140950957883559334686896904398903542182668795100841499722198631196089}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{17} - \frac{40039656885315819068720500442349673168279432900787997850039480439487218682}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{16} - \frac{375728289435102623863976393948532245623493413958618684834459284471735154234}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{15} - \frac{387749968253620071132206545326449530740036563514075519374448178483228924259}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{14} + \frac{74611753243635617908550065767066417080314957583498052344109698816194884288}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{13} + \frac{49166466510091388438587976812216610563034622842668506436940362869815191437}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{12} + \frac{18381110790917858129735602914920483850293290751395460743808089337027948228}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{11} + \frac{163916050442621278054932040030633944414701880009158527166420298386960104152}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{10} + \frac{34313944492004964797960322860916882241181520710712724078675569827221850516}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{9} - \frac{122436882823262138942735361014565183439791872641787733169034737852235056138}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{8} + \frac{8929864446132202934490158847248765712684418849741615490838281344350403778}{43128055089469944498813169906104317564780463611929263146870937593799649625} a^{7} - \frac{30121004496968491349141479685566689637353493517822215147011733017702162192}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{6} - \frac{249950972457545530320112634265695372808380141477229648765385264726767123717}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{5} - \frac{70644582634321975545688658717490221101845290913131976905026974419802653451}{163886609339985789095490045643196406746165761725331199958109562856438668575} a^{4} + \frac{8570505700972802760035818787430700722289814272625465799130114961597107526}{23412372762855112727927149377599486678023680246475885708301366122348381225} a^{3} - \frac{325169439665242239476464712577142560880070968190654032913352044861288236792}{819433046699928945477450228215982033730828808626655999790547814282193342875} a^{2} - \frac{30752974495533762605004162950027094189590314507847019955991779662422352627}{819433046699928945477450228215982033730828808626655999790547814282193342875} a - \frac{9172856040377971125194314964116507162293098414842208354518491253933856812}{117061863814275563639635746887997433390118401232379428541506830611741906125}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{13}) \), 13.1.330928743953809.1

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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ $26$ R ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}$

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
13Data not computed
263Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.263.2t1.a.a$1$ $ 263 $ \(\Q(\sqrt{-263}) \) $C_2$ (as 2T1) $1$ $-1$
1.3419.2t1.a.a$1$ $ 13 \cdot 263 $ \(\Q(\sqrt{-3419}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.44447.26t3.a.e$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.e$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.c$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.44447.26t3.a.c$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.44447.26t3.a.a$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.d$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.b$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.a$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.44447.26t3.a.b$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.44447.26t3.a.f$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.f$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.44447.26t3.a.d$2$ $ 13^{2} \cdot 263 $ 26.2.33169014017307639762747533311185137385697693.1 $D_{26}$ (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 *.