Properties

Label 25.1.397...641.1
Degree $25$
Signature $[1, 12]$
Discriminant $3.975\times 10^{43}$
Root discriminant $55.46$
Ramified primes $11, 79$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{25}$ (as 25T4)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 + 2*x^23 - 2*x^22 + 114*x^21 + 14*x^20 + 388*x^19 + 1085*x^18 + 7513*x^17 + 4819*x^16 + 1219*x^15 - 640*x^14 + 110085*x^13 + 396579*x^12 + 932255*x^11 + 3052375*x^10 + 5536096*x^9 + 12433767*x^8 + 17351055*x^7 + 27726417*x^6 + 30856797*x^5 + 36500635*x^4 + 30496620*x^3 + 26587890*x^2 + 14289723*x + 8362683)
 
gp: K = bnfinit(x^25 - x^24 + 2*x^23 - 2*x^22 + 114*x^21 + 14*x^20 + 388*x^19 + 1085*x^18 + 7513*x^17 + 4819*x^16 + 1219*x^15 - 640*x^14 + 110085*x^13 + 396579*x^12 + 932255*x^11 + 3052375*x^10 + 5536096*x^9 + 12433767*x^8 + 17351055*x^7 + 27726417*x^6 + 30856797*x^5 + 36500635*x^4 + 30496620*x^3 + 26587890*x^2 + 14289723*x + 8362683, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8362683, 14289723, 26587890, 30496620, 36500635, 30856797, 27726417, 17351055, 12433767, 5536096, 3052375, 932255, 396579, 110085, -640, 1219, 4819, 7513, 1085, 388, 14, 114, -2, 2, -1, 1]);
 

\(x^{25} - x^{24} + 2 x^{23} - 2 x^{22} + 114 x^{21} + 14 x^{20} + 388 x^{19} + 1085 x^{18} + 7513 x^{17} + 4819 x^{16} + 1219 x^{15} - 640 x^{14} + 110085 x^{13} + 396579 x^{12} + 932255 x^{11} + 3052375 x^{10} + 5536096 x^{9} + 12433767 x^{8} + 17351055 x^{7} + 27726417 x^{6} + 30856797 x^{5} + 36500635 x^{4} + 30496620 x^{3} + 26587890 x^{2} + 14289723 x + 8362683\)  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:  \(39753813747116100568718553941184504222638641\)\(\medspace = 11^{20}\cdot 79^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $55.46$
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:  $11, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{63} a^{17} - \frac{1}{63} a^{16} - \frac{1}{7} a^{15} + \frac{2}{63} a^{14} - \frac{10}{63} a^{13} - \frac{5}{63} a^{12} + \frac{1}{7} a^{11} + \frac{1}{9} a^{10} - \frac{1}{63} a^{9} - \frac{2}{9} a^{8} + \frac{5}{21} a^{7} - \frac{2}{63} a^{6} - \frac{2}{63} a^{5} - \frac{16}{63} a^{4} - \frac{8}{21} a^{3} - \frac{1}{9} a^{2} - \frac{2}{21} a$, $\frac{1}{189} a^{18} + \frac{4}{189} a^{16} + \frac{4}{27} a^{15} + \frac{20}{189} a^{14} + \frac{1}{7} a^{13} - \frac{31}{189} a^{12} + \frac{2}{189} a^{11} + \frac{20}{189} a^{10} - \frac{5}{63} a^{9} - \frac{76}{189} a^{8} + \frac{83}{189} a^{7} + \frac{73}{189} a^{6} + \frac{29}{63} a^{5} - \frac{68}{189} a^{4} - \frac{59}{189} a^{3} - \frac{2}{63} a^{2} - \frac{10}{21} a$, $\frac{1}{189} a^{19} + \frac{1}{189} a^{17} + \frac{10}{189} a^{16} + \frac{26}{189} a^{15} - \frac{1}{9} a^{14} - \frac{1}{189} a^{13} - \frac{25}{189} a^{12} + \frac{2}{27} a^{11} + \frac{2}{63} a^{10} - \frac{10}{189} a^{9} - \frac{43}{189} a^{8} - \frac{2}{27} a^{7} + \frac{8}{21} a^{6} + \frac{1}{189} a^{5} + \frac{31}{189} a^{4} - \frac{3}{7} a^{3} - \frac{16}{63} a^{2} + \frac{3}{7} a$, $\frac{1}{189} a^{20} + \frac{1}{189} a^{17} + \frac{10}{189} a^{16} + \frac{11}{189} a^{15} - \frac{2}{21} a^{14} - \frac{25}{189} a^{13} - \frac{5}{63} a^{12} + \frac{1}{27} a^{11} + \frac{4}{63} a^{10} - \frac{19}{189} a^{9} + \frac{20}{189} a^{8} + \frac{1}{189} a^{7} + \frac{17}{63} a^{6} + \frac{88}{189} a^{5} + \frac{47}{189} a^{4} + \frac{17}{189} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{202419} a^{21} + \frac{338}{202419} a^{20} - \frac{277}{202419} a^{19} - \frac{283}{202419} a^{18} - \frac{1030}{202419} a^{17} - \frac{1388}{202419} a^{16} + \frac{2150}{22491} a^{15} + \frac{13330}{202419} a^{14} + \frac{23000}{202419} a^{13} + \frac{2836}{202419} a^{12} + \frac{3386}{28917} a^{11} - \frac{21932}{202419} a^{10} + \frac{67}{28917} a^{9} - \frac{92800}{202419} a^{8} + \frac{10684}{67473} a^{7} + \frac{47249}{202419} a^{6} + \frac{24953}{67473} a^{5} - \frac{5881}{67473} a^{4} + \frac{74411}{202419} a^{3} - \frac{19415}{67473} a^{2} - \frac{382}{3213} a + \frac{76}{153}$, $\frac{1}{202419} a^{22} + \frac{76}{202419} a^{20} + \frac{166}{202419} a^{19} + \frac{376}{202419} a^{18} - \frac{1}{153} a^{17} - \frac{2024}{202419} a^{16} - \frac{7793}{202419} a^{15} + \frac{3287}{22491} a^{14} - \frac{4637}{67473} a^{13} + \frac{520}{7497} a^{12} - \frac{5591}{67473} a^{11} + \frac{24656}{202419} a^{10} + \frac{5476}{67473} a^{9} - \frac{77167}{202419} a^{8} - \frac{38842}{202419} a^{7} - \frac{49858}{202419} a^{6} + \frac{28045}{67473} a^{5} - \frac{2221}{28917} a^{4} - \frac{13306}{28917} a^{3} - \frac{1087}{3969} a^{2} + \frac{356}{3213} a + \frac{16}{153}$, $\frac{1}{729530018721} a^{23} - \frac{95852}{729530018721} a^{22} + \frac{261827}{243176672907} a^{21} + \frac{459687686}{243176672907} a^{20} - \frac{284785286}{243176672907} a^{19} + \frac{1306593212}{729530018721} a^{18} - \frac{5611969978}{729530018721} a^{17} - \frac{25219645763}{729530018721} a^{16} - \frac{108810494885}{729530018721} a^{15} - \frac{538356223}{14888367729} a^{14} - \frac{114150653060}{729530018721} a^{13} + \frac{97799404604}{729530018721} a^{12} - \frac{1095833686}{27019630323} a^{11} + \frac{92964379549}{729530018721} a^{10} - \frac{114494775386}{729530018721} a^{9} - \frac{71001930508}{729530018721} a^{8} + \frac{177466984888}{729530018721} a^{7} + \frac{80485093946}{243176672907} a^{6} + \frac{225045443864}{729530018721} a^{5} - \frac{340496355662}{729530018721} a^{4} + \frac{67766877001}{243176672907} a^{3} - \frac{4022517235}{11579841567} a^{2} - \frac{1347646736}{3859947189} a - \frac{90407237}{183807009}$, $\frac{1}{22701008026988302845353028058426337381210144897084363647721451} a^{24} + \frac{2227309950776986035263605650968560922563330415620}{7567002675662767615117676019475445793736714965694787882573817} a^{23} - \frac{7638207889952184524300567410058175476718944621006198887}{22701008026988302845353028058426337381210144897084363647721451} a^{22} + \frac{1455829141520661445417211878018658934493738828532469242}{840778075073640846124186224386160643748523885077198653619313} a^{21} + \frac{18777690818304022681250940231453035765195833904569283628164}{7567002675662767615117676019475445793736714965694787882573817} a^{20} - \frac{4669081724735466160634067560575762608225199725706567619428}{3243001146712614692193289722632333911601449271012051949674493} a^{19} + \frac{278945489707811991365841026952643216219111432045268355446}{840778075073640846124186224386160643748523885077198653619313} a^{18} - \frac{9343349676302418752154938454916472596967873697778296431656}{3243001146712614692193289722632333911601449271012051949674493} a^{17} + \frac{58167672634471352137502043651264769672800551761875454111}{10379976235477047483014644745508156095660788704656773501473} a^{16} + \frac{3360320749355728666240254624292618670092672004403969168243736}{22701008026988302845353028058426337381210144897084363647721451} a^{15} + \frac{691936626657289211499463990220185553197199756418580213719536}{22701008026988302845353028058426337381210144897084363647721451} a^{14} - \frac{787083829078255593964823735211495642889526147439735519568358}{22701008026988302845353028058426337381210144897084363647721451} a^{13} - \frac{1622996052989923004096742506904704170820539730441779551154655}{22701008026988302845353028058426337381210144897084363647721451} a^{12} + \frac{3036023680356415038355741078411906238545155465530510129709655}{22701008026988302845353028058426337381210144897084363647721451} a^{11} + \frac{6196898360824490776111120198175276077665194305471684175677}{445117804450751036183392707027967399631571468570281640151401} a^{10} - \frac{193654330796123393979282299025215476254881450950074933814640}{22701008026988302845353028058426337381210144897084363647721451} a^{9} + \frac{3670582416894762422993357029064135905427235131640977020219664}{22701008026988302845353028058426337381210144897084363647721451} a^{8} - \frac{3958433503053640983241022435707799530567102784531792330158528}{22701008026988302845353028058426337381210144897084363647721451} a^{7} - \frac{10526161353958727226376705534567653618738166207020284126988358}{22701008026988302845353028058426337381210144897084363647721451} a^{6} + \frac{2479535372808663269079122226577830132783617701615687437832073}{22701008026988302845353028058426337381210144897084363647721451} a^{5} + \frac{2090825038839505210267161768133634803075983344398218556700765}{22701008026988302845353028058426337381210144897084363647721451} a^{4} - \frac{2604712683121343802495183845552193509989542992045584055833741}{7567002675662767615117676019475445793736714965694787882573817} a^{3} + \frac{103813157976241702241574408889769971308750731562381942794740}{360333460745846076910365524736925990177938807890227994408277} a^{2} - \frac{1047155043791800327797212792398442118817881552801365944726}{17158736225992670329065024987472666198949467042391809257537} a - \frac{8327459516615307999790756319843158924231358287802864311}{22083315606168172881679568838446159844207808291366549881}$  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:  \( 508763901267.4053 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 508763901267.4053 \cdot 5}{2\sqrt{39753813747116100568718553941184504222638641}}\approx 1.52740889068023$ (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

5.1.6241.1

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 $25$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $25$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.79.2t1.a.a$1$ $ 79 $ \(\Q(\sqrt{-79}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.79.5t2.a.a$2$ $ 79 $ 5.1.6241.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.79.5t2.a.b$2$ $ 79 $ 5.1.6241.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.9559.25t4.a.d$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.f$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.g$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.a$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.b$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.h$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.c$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.e$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.j$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.9559.25t4.a.i$2$ $ 11^{2} \cdot 79 $ 25.1.39753813747116100568718553941184504222638641.1 $D_{25}$ (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 *.