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Properties

Label 25.25.9637561175...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 71^{20}$
Root discriminant $397.52$
Ramified primes $5, 71$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134965843, -84655405, -7540471425, 10679386965, 26925541130, -28270808806, -47424572900, 23961479550, 45375238060, -2960488270, -21126485889, -4248220395, 4092190190, 1439989960, -368334245, -196009889, 13892640, 14152650, 57555, -584805, -20596, 13920, 570, -180, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 180*x^23 + 570*x^22 + 13920*x^21 - 20596*x^20 - 584805*x^19 + 57555*x^18 + 14152650*x^17 + 13892640*x^16 - 196009889*x^15 - 368334245*x^14 + 1439989960*x^13 + 4092190190*x^12 - 4248220395*x^11 - 21126485889*x^10 - 2960488270*x^9 + 45375238060*x^8 + 23961479550*x^7 - 47424572900*x^6 - 28270808806*x^5 + 26925541130*x^4 + 10679386965*x^3 - 7540471425*x^2 - 84655405*x + 134965843)
 
gp: K = bnfinit(x^25 - 5*x^24 - 180*x^23 + 570*x^22 + 13920*x^21 - 20596*x^20 - 584805*x^19 + 57555*x^18 + 14152650*x^17 + 13892640*x^16 - 196009889*x^15 - 368334245*x^14 + 1439989960*x^13 + 4092190190*x^12 - 4248220395*x^11 - 21126485889*x^10 - 2960488270*x^9 + 45375238060*x^8 + 23961479550*x^7 - 47424572900*x^6 - 28270808806*x^5 + 26925541130*x^4 + 10679386965*x^3 - 7540471425*x^2 - 84655405*x + 134965843, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 180 x^{23} + 570 x^{22} + 13920 x^{21} - 20596 x^{20} - 584805 x^{19} + 57555 x^{18} + 14152650 x^{17} + 13892640 x^{16} - 196009889 x^{15} - 368334245 x^{14} + 1439989960 x^{13} + 4092190190 x^{12} - 4248220395 x^{11} - 21126485889 x^{10} - 2960488270 x^{9} + 45375238060 x^{8} + 23961479550 x^{7} - 47424572900 x^{6} - 28270808806 x^{5} + 26925541130 x^{4} + 10679386965 x^{3} - 7540471425 x^{2} - 84655405 x + 134965843 \)

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

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(96375611759789726717489130716177918666289770044386386871337890625=5^{40}\cdot 71^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $397.52$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1775=5^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1775}(1,·)$, $\chi_{1775}(451,·)$, $\chi_{1775}(196,·)$, $\chi_{1775}(711,·)$, $\chi_{1775}(1161,·)$, $\chi_{1775}(906,·)$, $\chi_{1775}(76,·)$, $\chi_{1775}(1421,·)$, $\chi_{1775}(1616,·)$, $\chi_{1775}(786,·)$, $\chi_{1775}(341,·)$, $\chi_{1775}(1496,·)$, $\chi_{1775}(1051,·)$, $\chi_{1775}(96,·)$, $\chi_{1775}(1761,·)$, $\chi_{1775}(356,·)$, $\chi_{1775}(806,·)$, $\chi_{1775}(551,·)$, $\chi_{1775}(1066,·)$, $\chi_{1775}(1516,·)$, $\chi_{1775}(1261,·)$, $\chi_{1775}(431,·)$, $\chi_{1775}(1141,·)$, $\chi_{1775}(696,·)$, $\chi_{1775}(1406,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{322} a^{20} + \frac{29}{161} a^{19} + \frac{15}{322} a^{18} - \frac{5}{161} a^{17} - \frac{2}{23} a^{16} + \frac{31}{161} a^{15} + \frac{9}{161} a^{14} - \frac{12}{161} a^{12} - \frac{39}{161} a^{11} - \frac{107}{322} a^{10} - \frac{137}{322} a^{9} + \frac{71}{322} a^{8} + \frac{7}{46} a^{7} + \frac{1}{23} a^{6} - \frac{17}{161} a^{5} + \frac{58}{161} a^{4} + \frac{31}{322} a^{3} + \frac{1}{46} a^{2} - \frac{89}{322} a - \frac{121}{322}$, $\frac{1}{322} a^{21} + \frac{16}{161} a^{19} - \frac{75}{322} a^{18} + \frac{3}{14} a^{17} + \frac{38}{161} a^{16} - \frac{18}{161} a^{15} - \frac{39}{161} a^{14} - \frac{12}{161} a^{13} + \frac{13}{161} a^{12} + \frac{5}{23} a^{11} + \frac{8}{23} a^{10} + \frac{64}{161} a^{9} + \frac{117}{322} a^{8} - \frac{13}{46} a^{7} - \frac{41}{322} a^{6} + \frac{78}{161} a^{5} + \frac{65}{322} a^{4} - \frac{10}{161} a^{3} + \frac{149}{322} a^{2} + \frac{25}{161} a - \frac{33}{161}$, $\frac{1}{65044} a^{22} + \frac{47}{32522} a^{21} + \frac{25}{65044} a^{20} - \frac{245}{4646} a^{19} - \frac{9179}{65044} a^{18} - \frac{3494}{16261} a^{17} + \frac{1182}{16261} a^{16} - \frac{2906}{16261} a^{15} - \frac{2434}{16261} a^{14} - \frac{2570}{16261} a^{13} + \frac{12503}{65044} a^{12} + \frac{333}{4646} a^{11} + \frac{27183}{65044} a^{10} + \frac{3173}{32522} a^{9} + \frac{697}{32522} a^{8} + \frac{10021}{32522} a^{7} + \frac{2754}{16261} a^{6} - \frac{12561}{32522} a^{5} + \frac{975}{4646} a^{4} - \frac{4673}{16261} a^{3} + \frac{21}{101} a^{2} + \frac{801}{4646} a - \frac{26287}{65044}$, $\frac{1}{130088} a^{23} - \frac{1}{130088} a^{22} + \frac{185}{130088} a^{21} + \frac{53}{130088} a^{20} + \frac{4177}{130088} a^{19} + \frac{3973}{130088} a^{18} - \frac{2215}{16261} a^{17} + \frac{247}{32522} a^{16} - \frac{555}{9292} a^{15} - \frac{3539}{32522} a^{14} + \frac{22937}{130088} a^{13} - \frac{14149}{130088} a^{12} + \frac{23845}{130088} a^{11} - \frac{5791}{130088} a^{10} + \frac{9837}{65044} a^{9} + \frac{142}{16261} a^{8} - \frac{7377}{16261} a^{7} + \frac{901}{32522} a^{6} - \frac{27939}{65044} a^{5} - \frac{5633}{16261} a^{4} + \frac{44}{16261} a^{3} - \frac{4119}{65044} a^{2} + \frac{1759}{18584} a - \frac{14807}{130088}$, $\frac{1}{1907687232594073327318899869789095638334595782000496781101662715753759529796958192968531433241068176} a^{24} - \frac{65611845779574846559417695702449956319033278109659715784810214052438999259235403344599925604}{119230452037129582957431241861818477395912236375031048818853919734609970612309887060533214577566761} a^{23} - \frac{397470350835959610054423390407050317470631437181777218809375221435237501559405475884715302988}{119230452037129582957431241861818477395912236375031048818853919734609970612309887060533214577566761} a^{22} + \frac{42926917919925150658906131782310016235852978855686766232975652787657601700159105716686327465723}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{21} - \frac{1253556083625763330603448846337196481946827357985328810172257203989831856807612491554770997856559}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{20} + \frac{1917057018375831538566133420594727123561649262636893088922956978534438890089107760974598990479139}{136263373756719523379921419270649688452471127285749770078690193982411394985497013783466530945790584} a^{19} + \frac{159969265831865503870437316611982282764338713300349477404644968691784758277314748950738193820812297}{1907687232594073327318899869789095638334595782000496781101662715753759529796958192968531433241068176} a^{18} + \frac{104578471790137822978248112529750862419527798547163171289621519151187314974123775028949723189703461}{476921808148518331829724967447273909583648945500124195275415678938439882449239548242132858310267044} a^{17} - \frac{229202058607768994482791554369337103479489665727042639736272749877102809468217528591459322512216145}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{16} - \frac{8988425600719352084262664717678324925962903565048592844744054907781461827678551119157272262189005}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{15} + \frac{369965050016984702936906388599933089308045582330980542024027316019571101106613881936554726135044581}{1907687232594073327318899869789095638334595782000496781101662715753759529796958192968531433241068176} a^{14} + \frac{104291140751899326227093626206974359690091684174113629801768666728601221442456116979026692233754453}{476921808148518331829724967447273909583648945500124195275415678938439882449239548242132858310267044} a^{13} + \frac{13835130148584221410371056485017905348498183504484470346995224435196056323846323966132138384834219}{238460904074259165914862483723636954791824472750062097637707839469219941224619774121066429155133522} a^{12} - \frac{22782570927550088648772962848097070857794552095897365151361313948376566603908333636222160499148587}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{11} - \frac{579609746229395856514548396042549341742701127192444618002851598563857556515452770062382178697070229}{1907687232594073327318899869789095638334595782000496781101662715753759529796958192968531433241068176} a^{10} + \frac{62580879064459226515230615955777482971094220314938522524657145377038252392511559893019304862915429}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{9} + \frac{46607543892100193794530561513611088435554916334017918152795676450417775360459760170826627074195535}{476921808148518331829724967447273909583648945500124195275415678938439882449239548242132858310267044} a^{8} + \frac{50539511570416540746352037460576323730558693536071016250260579462457252402201330328752279216388477}{119230452037129582957431241861818477395912236375031048818853919734609970612309887060533214577566761} a^{7} - \frac{193693521727704430199788186933241843809642330236185804379999741458770346139315506606501011299152801}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a^{6} + \frac{59096851911379831797041180691523046869570819756116383097171944384801584884346916494610072711348127}{136263373756719523379921419270649688452471127285749770078690193982411394985497013783466530945790584} a^{5} - \frac{120039075792792122073284733015123094262609309779697891931445127348234931882427213642590873611776179}{476921808148518331829724967447273909583648945500124195275415678938439882449239548242132858310267044} a^{4} - \frac{21747804561273253913190287304482955793689913426323672151149688383459301314594491009301684117937131}{136263373756719523379921419270649688452471127285749770078690193982411394985497013783466530945790584} a^{3} - \frac{9312264186057228827809014932363547285445427406736720931060220576542256884828775678450095047615489}{18887992401921518092266335344446491468659364178222740406947155601522371582148100920480509240010576} a^{2} + \frac{163506901627776308706639805191136626994250469355275865270505482680173042039726564174307640682375211}{953843616297036663659449934894547819167297891000248390550831357876879764898479096484265716620534088} a + \frac{693183851989500619326566958324849478023751583540936743117048008513234434311190446494297988411187}{272526747513439046759842838541299376904942254571499540157380387964822789970994027566933061891581168}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $24$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 206417156052773680000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.390625.1, 5.5.9926437890625.4, 5.5.9926437890625.2, 5.5.9926437890625.3, 5.5.25411681.1, 5.5.9926437890625.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
71Data not computed