Properties

Label 25.1.118...921.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.184\times 10^{46}$
Root discriminant $69.65$
Ramified primes $11, 127$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173)
 
gp: K = bnfinit(x^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1384173, 5156073, -12748972, -3014982, 4604688, 46691010, -82760080, 76250162, -51222985, 32382901, -9825280, 2807964, -456783, -336486, 158147, -40555, 14003, -17480, -1564, 1755, 1005, 156, -76, -19, -1, 1]);
 

\(x^{25} - x^{24} - 19 x^{23} - 76 x^{22} + 156 x^{21} + 1005 x^{20} + 1755 x^{19} - 1564 x^{18} - 17480 x^{17} + 14003 x^{16} - 40555 x^{15} + 158147 x^{14} - 336486 x^{13} - 456783 x^{12} + 2807964 x^{11} - 9825280 x^{10} + 32382901 x^{9} - 51222985 x^{8} + 76250162 x^{7} - 82760080 x^{6} + 46691010 x^{5} + 4604688 x^{4} - 3014982 x^{3} - 12748972 x^{2} + 5156073 x + 1384173\)  Toggle raw display

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(11843998797823466973474682749162211402125799921\)\(\medspace = 11^{20}\cdot 127^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.65$
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:  $11, 127$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{21} a^{16} + \frac{2}{21} a^{14} + \frac{1}{21} a^{13} + \frac{2}{21} a^{12} - \frac{2}{21} a^{11} + \frac{2}{21} a^{8} + \frac{1}{3} a^{6} - \frac{4}{21} a^{5} + \frac{10}{21} a^{4} - \frac{10}{21} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{63} a^{17} + \frac{1}{63} a^{16} - \frac{5}{63} a^{15} + \frac{10}{63} a^{14} + \frac{10}{63} a^{13} + \frac{1}{9} a^{12} - \frac{2}{63} a^{11} + \frac{1}{9} a^{10} + \frac{2}{63} a^{9} + \frac{2}{63} a^{8} - \frac{4}{9} a^{7} - \frac{25}{63} a^{6} + \frac{20}{63} a^{5} + \frac{2}{9} a^{4} - \frac{4}{63} a^{3} - \frac{31}{63} a^{2} - \frac{1}{7} a$, $\frac{1}{315} a^{18} - \frac{2}{315} a^{17} - \frac{1}{63} a^{16} - \frac{17}{315} a^{15} + \frac{4}{45} a^{14} + \frac{22}{315} a^{13} + \frac{4}{315} a^{12} + \frac{7}{45} a^{11} + \frac{2}{315} a^{10} - \frac{5}{63} a^{9} - \frac{4}{45} a^{8} + \frac{101}{315} a^{7} + \frac{74}{315} a^{6} - \frac{37}{315} a^{5} - \frac{37}{315} a^{4} - \frac{4}{45} a^{3} - \frac{1}{7} a^{2} + \frac{7}{15} a + \frac{2}{5}$, $\frac{1}{315} a^{19} + \frac{1}{315} a^{17} - \frac{2}{315} a^{16} + \frac{7}{45} a^{15} - \frac{2}{315} a^{14} - \frac{47}{315} a^{13} + \frac{52}{315} a^{12} + \frac{10}{63} a^{11} + \frac{7}{45} a^{10} + \frac{47}{315} a^{9} + \frac{19}{63} a^{8} - \frac{109}{315} a^{7} - \frac{139}{315} a^{6} - \frac{76}{315} a^{5} - \frac{22}{315} a^{4} + \frac{8}{105} a^{3} + \frac{152}{315} a^{2} + \frac{2}{7} a - \frac{1}{5}$, $\frac{1}{945} a^{20} - \frac{1}{945} a^{19} - \frac{1}{945} a^{18} + \frac{2}{315} a^{17} + \frac{1}{45} a^{16} - \frac{2}{45} a^{15} + \frac{23}{315} a^{14} - \frac{10}{63} a^{13} - \frac{13}{189} a^{12} + \frac{86}{945} a^{11} - \frac{76}{945} a^{10} + \frac{1}{315} a^{9} - \frac{76}{315} a^{8} - \frac{124}{315} a^{7} - \frac{1}{9} a^{6} - \frac{4}{315} a^{5} - \frac{52}{189} a^{4} - \frac{436}{945} a^{3} - \frac{292}{945} a^{2} + \frac{37}{105} a + \frac{2}{15}$, $\frac{1}{2835} a^{21} + \frac{1}{2835} a^{19} + \frac{2}{2835} a^{18} + \frac{1}{135} a^{17} + \frac{2}{315} a^{16} - \frac{22}{189} a^{15} + \frac{4}{135} a^{14} + \frac{418}{2835} a^{13} + \frac{10}{189} a^{12} - \frac{362}{2835} a^{11} - \frac{37}{2835} a^{10} + \frac{92}{945} a^{9} + \frac{86}{315} a^{8} - \frac{334}{945} a^{7} + \frac{188}{945} a^{6} + \frac{76}{2835} a^{5} - \frac{452}{945} a^{4} + \frac{298}{2835} a^{3} - \frac{523}{2835} a^{2} + \frac{8}{35} a - \frac{22}{45}$, $\frac{1}{2835} a^{22} + \frac{1}{2835} a^{20} + \frac{2}{2835} a^{19} + \frac{1}{945} a^{18} + \frac{1}{315} a^{17} - \frac{1}{189} a^{16} - \frac{22}{189} a^{15} + \frac{4}{2835} a^{14} - \frac{142}{945} a^{13} - \frac{209}{2835} a^{12} - \frac{424}{2835} a^{11} - \frac{5}{189} a^{10} + \frac{1}{15} a^{9} - \frac{16}{945} a^{8} + \frac{317}{945} a^{7} - \frac{1076}{2835} a^{6} + \frac{11}{189} a^{5} + \frac{199}{2835} a^{4} - \frac{649}{2835} a^{3} - \frac{19}{45} a^{2} + \frac{62}{315} a + \frac{1}{5}$, $\frac{1}{99225} a^{23} - \frac{1}{6615} a^{22} + \frac{8}{99225} a^{21} + \frac{2}{99225} a^{20} + \frac{64}{99225} a^{19} - \frac{82}{99225} a^{18} - \frac{148}{33075} a^{17} + \frac{556}{33075} a^{16} - \frac{269}{3969} a^{15} + \frac{418}{33075} a^{14} - \frac{8686}{99225} a^{13} - \frac{14026}{99225} a^{12} + \frac{1396}{99225} a^{11} - \frac{181}{2025} a^{10} + \frac{1144}{33075} a^{9} - \frac{5692}{33075} a^{8} - \frac{27836}{99225} a^{7} + \frac{9679}{33075} a^{6} - \frac{22723}{99225} a^{5} - \frac{20599}{99225} a^{4} - \frac{6844}{19845} a^{3} - \frac{5741}{19845} a^{2} - \frac{1441}{3675} a + \frac{677}{1575}$, $\frac{1}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{24} - \frac{114038051769016907003597104696069426180403623637213159984620212684488281517902}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{23} + \frac{1096801369275434024937853981645152665819844761354960461308917461537399967818201}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{22} - \frac{141884572453253625828486929481793855841066321217670658232720927254114006326049}{1509491888474598078593201098364666714768798585793439361770546515524094294670016625} a^{21} - \frac{1001720635871612617634988994055977219291149221612311431087697225382684705543212}{2113288643864437310030481537710533400676318020110815106478765121733732012538023275} a^{20} + \frac{83466462083233775378695182875508104006468584067069351489017025600942237040559}{60379675538983923143728043934586668590751943431737574470821860620963771786800665} a^{19} - \frac{436531739746908728232344476102737761106355700682234769562705579997026729664394}{905695133084758847155920659018800028861279151476063617062327909314456576802009975} a^{18} + \frac{67112406102248529182802296705687736765124607661480789504065823703312131810290687}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{17} - \frac{362795515572153725574676067766853179965908117962682891483640293797203276914218896}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{16} - \frac{2619217938397126706193822228648252997943498618336118908095496169671796714608810191}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{15} + \frac{581887380683782930852000233810839543018233743852164934162712403793719312519961962}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{14} + \frac{44613118863561986303415054838666671581392842597365143604119681380571020269664196}{1509491888474598078593201098364666714768798585793439361770546515524094294670016625} a^{13} - \frac{238037889710868554591280160730436127835104842991113278440147122771973032192269149}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{12} + \frac{21764497158157458569695077322634669184809404588545529586056376115388147223316473}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{11} + \frac{146352066842279714126086990342174826993102677331049785493761485291835255323788593}{6339865931593311930091444613131600202028954060332445319436295365201196037614069825} a^{10} - \frac{5844104571805726318853200079792419255423778132368431442014798201806760517745888}{60379675538983923143728043934586668590751943431737574470821860620963771786800665} a^{9} - \frac{14711729059731392032487599114095057779883212018158164258358014986675721414884138909}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{8} + \frac{326644600159037772040900512175827293108705226812503154675542189413120087915626401}{1093080333033329643119214588470965552073957596609042296454533683655378627174839625} a^{7} - \frac{375487488537515340547817665443228574746541563259005233506143245708016249280228343}{3522147739774062183384135896184222334460530033518025177464608536222886687563372125} a^{6} - \frac{3701124123645145901987597482819852290580638889906861527781620163768978242505687001}{10566443219322186550152407688552667003381590100554075532393825608668660062690116375} a^{5} - \frac{587030055661696122366780576869103587419806023415608547415166336629786447443330658}{3522147739774062183384135896184222334460530033518025177464608536222886687563372125} a^{4} - \frac{40865886478882511379837858078629980172794838409189725291064844025466122033525867}{234809849318270812225609059745614822297368668901201678497640569081525779170891475} a^{3} - \frac{738948698312638487157725146919395992501776507129115715375531390173860488806729287}{31699329657966559650457223065658001010144770301662226597181476826005980188070349125} a^{2} + \frac{105329329135378258872114272047569892114300010652028853408912800812374596488776252}{704429547954812436676827179236844466892106006703605035492921707244577337512674425} a + \frac{224799166070650145047364619736808923067092649479685029477350587142834931101508561}{503163962824866026197733699454888904922932861931146453923515505174698098223338875}$  Toggle raw display

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

Class group and class number

$C_{5}$, which has order $5$ (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 $2$)  Toggle raw display
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:  \( 14739079523494.871 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 14739079523494.871 \cdot 5}{2\sqrt{11843998797823466973474682749162211402125799921}}\approx 2.56359614224321$ (assuming GRH)

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.16129.1

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 $25$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $25$ $25$ $25$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $25$ $25$ $25$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
11Data not computed
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.127.2t1.a.a$1$ $ 127 $ \(\Q(\sqrt{-127}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.127.5t2.a.a$2$ $ 127 $ 5.1.16129.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.127.5t2.a.b$2$ $ 127 $ 5.1.16129.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.15367.25t4.a.b$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.i$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.e$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.g$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.a$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.j$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.h$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.f$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.d$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.15367.25t4.a.c$2$ $ 11^{2} \cdot 127 $ 25.1.11843998797823466973474682749162211402125799921.1 $D_{25}$ (as 25T4) $1$ $0$

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 *.