Properties

Label 26.2.144...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $1.447\times 10^{43}$
Root discriminant $45.71$
Ramified primes $5, 691$
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 - 3*x^25 + 8*x^24 - 23*x^23 - 44*x^22 + 17*x^21 + 450*x^20 - 860*x^19 + 3401*x^18 + 4044*x^17 - 12200*x^16 - 9996*x^15 + 5376*x^14 - 176112*x^13 - 67837*x^12 - 1501*x^11 - 261354*x^10 - 143521*x^9 - 1171*x^8 - 152170*x^7 - 30780*x^6 - 39875*x^5 - 16350*x^4 - 26625*x^3 - 13000*x^2 - 6875*x - 3125)
 
gp: K = bnfinit(x^26 - 3*x^25 + 8*x^24 - 23*x^23 - 44*x^22 + 17*x^21 + 450*x^20 - 860*x^19 + 3401*x^18 + 4044*x^17 - 12200*x^16 - 9996*x^15 + 5376*x^14 - 176112*x^13 - 67837*x^12 - 1501*x^11 - 261354*x^10 - 143521*x^9 - 1171*x^8 - 152170*x^7 - 30780*x^6 - 39875*x^5 - 16350*x^4 - 26625*x^3 - 13000*x^2 - 6875*x - 3125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3125, -6875, -13000, -26625, -16350, -39875, -30780, -152170, -1171, -143521, -261354, -1501, -67837, -176112, 5376, -9996, -12200, 4044, 3401, -860, 450, 17, -44, -23, 8, -3, 1]);
 

\( x^{26} - 3 x^{25} + 8 x^{24} - 23 x^{23} - 44 x^{22} + 17 x^{21} + 450 x^{20} - 860 x^{19} + 3401 x^{18} + 4044 x^{17} - 12200 x^{16} - 9996 x^{15} + 5376 x^{14} - 176112 x^{13} - 67837 x^{12} - 1501 x^{11} - 261354 x^{10} - 143521 x^{9} - 1171 x^{8} - 152170 x^{7} - 30780 x^{6} - 39875 x^{5} - 16350 x^{4} - 26625 x^{3} - 13000 x^{2} - 6875 x - 3125 \)

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:  \(14465938847694062481283347921338966064453125\)\(\medspace = 5^{13}\cdot 691^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.71$
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:  $5, 691$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{11} + \frac{1}{25} a^{9} - \frac{2}{25} a^{8} - \frac{9}{25} a^{7} - \frac{6}{25} a^{6} - \frac{11}{25} a^{5} - \frac{3}{25} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{12} + \frac{1}{25} a^{10} - \frac{2}{25} a^{9} + \frac{1}{25} a^{8} + \frac{9}{25} a^{7} - \frac{6}{25} a^{6} - \frac{3}{25} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{13} + \frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{4}{25} a^{7} - \frac{8}{25} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} + \frac{2}{25} a^{8} - \frac{2}{25} a^{7} + \frac{4}{25} a^{6} - \frac{6}{25} a^{5} + \frac{7}{25} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{13} + \frac{2}{25} a^{12} + \frac{1}{25} a^{11} + \frac{2}{25} a^{9} - \frac{2}{25} a^{8} + \frac{4}{25} a^{7} - \frac{6}{25} a^{6} + \frac{7}{25} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{125} a^{19} + \frac{2}{125} a^{18} + \frac{2}{125} a^{17} + \frac{1}{125} a^{16} - \frac{1}{125} a^{15} - \frac{1}{125} a^{14} - \frac{12}{125} a^{13} - \frac{2}{125} a^{12} - \frac{11}{125} a^{11} - \frac{9}{125} a^{10} + \frac{3}{125} a^{9} + \frac{6}{125} a^{8} + \frac{61}{125} a^{7} - \frac{17}{125} a^{6} - \frac{18}{125} a^{5} + \frac{4}{25} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{125} a^{20} - \frac{2}{125} a^{18} + \frac{2}{125} a^{17} + \frac{2}{125} a^{16} + \frac{1}{125} a^{15} - \frac{8}{125} a^{13} - \frac{2}{125} a^{12} - \frac{7}{125} a^{11} - \frac{9}{125} a^{10} - \frac{2}{25} a^{9} + \frac{9}{125} a^{8} - \frac{44}{125} a^{7} + \frac{61}{125} a^{6} + \frac{16}{125} a^{5} + \frac{8}{25} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{21} + \frac{1}{125} a^{18} + \frac{1}{125} a^{17} - \frac{2}{125} a^{16} - \frac{2}{125} a^{15} - \frac{1}{125} a^{13} - \frac{1}{125} a^{12} - \frac{11}{125} a^{11} + \frac{2}{125} a^{10} + \frac{2}{25} a^{9} + \frac{3}{125} a^{8} - \frac{37}{125} a^{7} - \frac{3}{125} a^{6} - \frac{61}{125} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{125} a^{22} - \frac{1}{125} a^{18} + \frac{1}{125} a^{17} + \frac{2}{125} a^{16} + \frac{1}{125} a^{15} + \frac{6}{125} a^{13} - \frac{4}{125} a^{12} + \frac{3}{125} a^{11} - \frac{11}{125} a^{10} + \frac{1}{25} a^{9} + \frac{12}{125} a^{8} - \frac{54}{125} a^{7} + \frac{11}{125} a^{6} - \frac{37}{125} a^{5} - \frac{12}{25} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{6875} a^{23} - \frac{9}{6875} a^{22} - \frac{27}{6875} a^{21} - \frac{1}{275} a^{20} - \frac{26}{6875} a^{19} + \frac{18}{6875} a^{18} - \frac{4}{6875} a^{17} - \frac{118}{6875} a^{16} - \frac{18}{1375} a^{15} + \frac{81}{6875} a^{14} - \frac{641}{6875} a^{13} - \frac{34}{625} a^{12} - \frac{76}{6875} a^{11} + \frac{1}{1375} a^{10} - \frac{493}{6875} a^{9} + \frac{542}{6875} a^{8} - \frac{3134}{6875} a^{7} - \frac{679}{1375} a^{6} - \frac{222}{1375} a^{5} + \frac{131}{275} a^{4} - \frac{101}{275} a^{3} + \frac{27}{55} a^{2} - \frac{18}{55} a - \frac{2}{11}$, $\frac{1}{108521875} a^{24} + \frac{2171}{108521875} a^{23} + \frac{40688}{108521875} a^{22} + \frac{64299}{21704375} a^{21} + \frac{155684}{108521875} a^{20} - \frac{9687}{2646875} a^{19} - \frac{438549}{108521875} a^{18} - \frac{2009463}{108521875} a^{17} - \frac{141446}{21704375} a^{16} - \frac{12139}{9865625} a^{15} - \frac{324781}{108521875} a^{14} + \frac{1862206}{108521875} a^{13} - \frac{82393}{15503125} a^{12} - \frac{878408}{21704375} a^{11} + \frac{2865512}{108521875} a^{10} - \frac{219938}{108521875} a^{9} + \frac{4478}{15503125} a^{8} - \frac{2788897}{21704375} a^{7} - \frac{8494229}{21704375} a^{6} + \frac{9887}{124025} a^{5} - \frac{1364576}{4340875} a^{4} - \frac{97326}{868175} a^{3} + \frac{113534}{868175} a^{2} - \frac{8192}{34727} a + \frac{7048}{34727}$, $\frac{1}{1343189559172320658256872377629851533505228144390625} a^{25} + \frac{230795201160586694574276921257987938814702}{191884222738902951179553196804264504786461163484375} a^{24} - \frac{1611076095964132728690084721810674341692220307}{28578501258985545920358986758081947521387832859375} a^{23} + \frac{533941532994796461211034077266419797558442503284}{1343189559172320658256872377629851533505228144390625} a^{22} - \frac{72012661106938617848998611986099091639315364776}{32760720955422455079435911649508573987932393765625} a^{21} - \frac{2114294438263175341915597089080285055597864232}{1204654313159031980499437109981929626462088021875} a^{20} + \frac{780088577366785619563704888418005563611941963093}{268637911834464131651374475525970306701045628878125} a^{19} - \frac{5109848743001855450403583299609780201311312590891}{268637911834464131651374475525970306701045628878125} a^{18} + \frac{8090166375870071298554474891421243563046459619416}{1343189559172320658256872377629851533505228144390625} a^{17} + \frac{2783427256163318361013998419212194761570897705391}{1343189559172320658256872377629851533505228144390625} a^{16} + \frac{17549883486311101113876859529547999839555626827897}{1343189559172320658256872377629851533505228144390625} a^{15} - \frac{25709571815193243675127399807442820776486993490397}{1343189559172320658256872377629851533505228144390625} a^{14} - \frac{869590457785592532235660043353315622570839260268}{122108141742938241659715670693622866682293467671875} a^{13} - \frac{18030669876960582090131783225914397078854424761153}{1343189559172320658256872377629851533505228144390625} a^{12} + \frac{369923626300618511131777008679238184524903562807}{32760720955422455079435911649508573987932393765625} a^{11} - \frac{48083868589532397896198272998141875441030562117497}{1343189559172320658256872377629851533505228144390625} a^{10} - \frac{114801771721232001208927781664431625572712486683253}{1343189559172320658256872377629851533505228144390625} a^{9} + \frac{14493312251722208285477015506661054378170956245753}{1343189559172320658256872377629851533505228144390625} a^{8} - \frac{67619662184260098407222035479021718304213272380839}{268637911834464131651374475525970306701045628878125} a^{7} - \frac{96175603857636588525566226412539097462937623450692}{268637911834464131651374475525970306701045628878125} a^{6} - \frac{627959817499965386536503505762613850472510082424}{53727582366892826330274895105194061340209125775625} a^{5} - \frac{1848867439073206805422511917499304411456877114188}{53727582366892826330274895105194061340209125775625} a^{4} - \frac{160330054467976588229853969376471909883839652202}{429820658935142610642199160841552490721673006205} a^{3} + \frac{4630971035353868166546285032856867336116789313982}{10745516473378565266054979021038812268041825155125} a^{2} - \frac{175817373441920883069308839443743415921941338027}{2149103294675713053210995804207762453608365031025} a - \frac{202101960699905096490776060269292610449153604417}{429820658935142610642199160841552490721673006205}$

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:  $13$
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:  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:  \( 2016100861553.7466 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 2016100861553.7466 \cdot 1}{2\sqrt{14465938847694062481283347921338966064453125}}\approx 4.01353781085479$ (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

\(\Q(\sqrt{5}) \), 13.1.1700937320873056890625.1

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

Sibling fields

Degree 26 sibling: 26.0.1999192748751319434913358682729045110107421875.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
691Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.691.2t1.a.a$1$ $ 691 $ \(\Q(\sqrt{-691}) \) $C_2$ (as 2T1) $1$ $-1$
1.3455.2t1.a.a$1$ $ 5 \cdot 691 $ \(\Q(\sqrt{-3455}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.3455.26t3.b.f$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.b$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.d$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.b.a$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.26t3.b.d$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.f$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.a$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.e$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.b.e$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.26t3.b.b$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.c$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.b.c$2$ $ 5 \cdot 691 $ 26.2.14465938847694062481283347921338966064453125.1 $D_{26}$ (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 *.