Properties

Label 26.0.549...875.1
Degree $26$
Signature $[0, 13]$
Discriminant $-5.496\times 10^{38}$
Root discriminant $30.90$
Ramified primes $5, 191$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 + 5*x^24 - 26*x^23 + 53*x^22 - 54*x^21 + 268*x^20 - 237*x^19 + 744*x^18 + 440*x^17 + 1338*x^16 - 3330*x^15 - 8756*x^14 + 7284*x^13 + 63381*x^12 + 167449*x^11 + 262693*x^10 + 320455*x^9 + 419505*x^8 + 480606*x^7 + 569660*x^6 + 459537*x^5 + 397335*x^4 + 258066*x^3 + 85342*x^2 + 34263*x + 39371)
 
gp: K = bnfinit(x^26 - 2*x^25 + 5*x^24 - 26*x^23 + 53*x^22 - 54*x^21 + 268*x^20 - 237*x^19 + 744*x^18 + 440*x^17 + 1338*x^16 - 3330*x^15 - 8756*x^14 + 7284*x^13 + 63381*x^12 + 167449*x^11 + 262693*x^10 + 320455*x^9 + 419505*x^8 + 480606*x^7 + 569660*x^6 + 459537*x^5 + 397335*x^4 + 258066*x^3 + 85342*x^2 + 34263*x + 39371, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39371, 34263, 85342, 258066, 397335, 459537, 569660, 480606, 419505, 320455, 262693, 167449, 63381, 7284, -8756, -3330, 1338, 440, 744, -237, 268, -54, 53, -26, 5, -2, 1]);
 

\( x^{26} - 2 x^{25} + 5 x^{24} - 26 x^{23} + 53 x^{22} - 54 x^{21} + 268 x^{20} - 237 x^{19} + 744 x^{18} + 440 x^{17} + 1338 x^{16} - 3330 x^{15} - 8756 x^{14} + 7284 x^{13} + 63381 x^{12} + 167449 x^{11} + 262693 x^{10} + 320455 x^{9} + 419505 x^{8} + 480606 x^{7} + 569660 x^{6} + 459537 x^{5} + 397335 x^{4} + 258066 x^{3} + 85342 x^{2} + 34263 x + 39371 \)

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:  \(-549596338332815511233443533045654296875\)\(\medspace = -\,5^{13}\cdot 191^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $30.90$
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:  $5, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{3}{7} a^{15} + \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{19} + \frac{2}{7} a^{16} + \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{20} - \frac{2}{7} a^{16} - \frac{3}{7} a^{15} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{21} + \frac{2}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{22} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} + \frac{3}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{133} a^{23} + \frac{6}{133} a^{22} - \frac{1}{19} a^{21} + \frac{8}{133} a^{20} + \frac{1}{19} a^{19} + \frac{1}{133} a^{18} - \frac{2}{133} a^{17} + \frac{9}{19} a^{16} - \frac{25}{133} a^{15} + \frac{44}{133} a^{14} - \frac{2}{7} a^{13} - \frac{44}{133} a^{12} - \frac{24}{133} a^{11} + \frac{61}{133} a^{10} - \frac{53}{133} a^{9} + \frac{33}{133} a^{8} - \frac{39}{133} a^{7} + \frac{22}{133} a^{6} - \frac{53}{133} a^{5} - \frac{65}{133} a^{4} - \frac{54}{133} a^{3} - \frac{7}{19} a^{2} - \frac{23}{133} a + \frac{48}{133}$, $\frac{1}{296989} a^{24} + \frac{695}{296989} a^{23} + \frac{1790}{296989} a^{22} - \frac{18495}{296989} a^{21} - \frac{4513}{296989} a^{20} + \frac{14837}{296989} a^{19} - \frac{682}{26999} a^{18} - \frac{821}{296989} a^{17} - \frac{736}{296989} a^{16} + \frac{81505}{296989} a^{15} - \frac{112108}{296989} a^{14} - \frac{1146}{3857} a^{13} + \frac{128405}{296989} a^{12} - \frac{1541}{296989} a^{11} - \frac{79567}{296989} a^{10} - \frac{139198}{296989} a^{9} + \frac{24294}{296989} a^{8} + \frac{15710}{42427} a^{7} - \frac{1427}{15631} a^{6} + \frac{114981}{296989} a^{5} + \frac{119929}{296989} a^{4} + \frac{34546}{296989} a^{3} - \frac{2454}{42427} a^{2} + \frac{41429}{296989} a - \frac{131905}{296989}$, $\frac{1}{2545501423375695016185898442997109704150424866063925872389941612703} a^{25} - \frac{180348242156501654060054639551044292086575306184886835755940}{133973759125036579799257812789321563376338150845469782757365348037} a^{24} + \frac{8269043277925647274182074031757594168010346764788512780806187025}{2545501423375695016185898442997109704150424866063925872389941612703} a^{23} + \frac{3612980444787427300540454325506783555565704483845516511115590219}{231409220306881365107808949363373609468220442369447806580903782973} a^{22} - \frac{181300680081499360126846703353937638080517585493579417728530658841}{2545501423375695016185898442997109704150424866063925872389941612703} a^{21} + \frac{714825761668454692626700875651924412241344774554269906332513984}{14063543775556326056275682005508893393096269978253734101601887363} a^{20} - \frac{51439414308027486651411186864089335717248881784549861323257266713}{2545501423375695016185898442997109704150424866063925872389941612703} a^{19} + \frac{134145359599336203488854038017302105221390193884161951561466287417}{2545501423375695016185898442997109704150424866063925872389941612703} a^{18} + \frac{172385548414260756009004257151522247520004618475514900826818733124}{2545501423375695016185898442997109704150424866063925872389941612703} a^{17} + \frac{1136323309940362885376345221995384984538204646178341583688847894217}{2545501423375695016185898442997109704150424866063925872389941612703} a^{16} + \frac{1200792454719719385974229336581934709820985469086300638484713294472}{2545501423375695016185898442997109704150424866063925872389941612703} a^{15} + \frac{160623921872768924081903165313201825500579590154238708833975630405}{363643060482242145169414063285301386307203552294846553198563087529} a^{14} - \frac{9340531479309017165753842155371586956123176564772195311586167631}{41729531530749098625998335131100159084433194525638129055572813323} a^{13} - \frac{57736561084479229724226415463973170324222637937678548061513647870}{231409220306881365107808949363373609468220442369447806580903782973} a^{12} - \frac{139935068520181076488524515157177879509651521121238292237654669527}{2545501423375695016185898442997109704150424866063925872389941612703} a^{11} - \frac{578760572616831608550727991063668929962420275239362170853801657597}{2545501423375695016185898442997109704150424866063925872389941612703} a^{10} - \frac{969121407057987738990725070165703563634757495295292652696173081996}{2545501423375695016185898442997109704150424866063925872389941612703} a^{9} - \frac{119229097867858184086610296875430291096955947335068562290288634668}{363643060482242145169414063285301386307203552294846553198563087529} a^{8} + \frac{313672749104280090463369790009044548725543450560218820819709243745}{2545501423375695016185898442997109704150424866063925872389941612703} a^{7} + \frac{40449904998535457031116210850157299570438582975173650620006818787}{2545501423375695016185898442997109704150424866063925872389941612703} a^{6} + \frac{26444689517178321351895194936292421519541646412313566650222145200}{62085400570138902833802401048709992784156704050339655424144917383} a^{5} - \frac{922309525163526691269518099844287757002085678696261866327514643236}{2545501423375695016185898442997109704150424866063925872389941612703} a^{4} - \frac{122320212441963456626671477335519965385884232749616147269096328806}{363643060482242145169414063285301386307203552294846553198563087529} a^{3} - \frac{414487829758034955697837497652694660246422726132175764875174269116}{2545501423375695016185898442997109704150424866063925872389941612703} a^{2} - \frac{1180555810115311122665960143926583870829647995823849102966864543193}{2545501423375695016185898442997109704150424866063925872389941612703} a + \frac{160979557151380438370578867736342759525419636286072196027691966225}{363643060482242145169414063285301386307203552294846553198563087529}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -1 \) (order $2$)
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:  \( 130484336.16997853 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 130484336.16997853 \cdot 4}{2\sqrt{549596338332815511233443533045654296875}}\approx 0.264791932686390$ (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

\(\Q(\sqrt{-955}) \), 13.1.48551226272641.1

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

Sibling fields

Degree 26 sibling: 26.2.2877467739962384875567767188720703125.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ $26$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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

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
5Data not computed
$191$191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.955.2t1.a.a$1$ $ 5 \cdot 191 $ \(\Q(\sqrt{-955}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.4775.26t3.a.e$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.a.d$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.a.c$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.a.f$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.a.a$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.a.b$2$ $ 5^{2} \cdot 191 $ 26.0.549596338332815511233443533045654296875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$

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 *.