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Properties

Label 25.25.4519470748...0801.1
Degree $25$
Signature $[25, 0]$
Discriminant $11^{20}\cdot 31^{20}$
Root discriminant $106.22$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group Trivial (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![51424, 718016, 3356864, 4451908, -10040832, -31826729, -6485584, 57112611, 46080424, -37286821, -52524876, 5785190, 27115460, 3886133, -7477592, -2099287, 1150294, 441736, -96798, -48221, 4072, 2826, -66, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 84*x^23 - 66*x^22 + 2826*x^21 + 4072*x^20 - 48221*x^19 - 96798*x^18 + 441736*x^17 + 1150294*x^16 - 2099287*x^15 - 7477592*x^14 + 3886133*x^13 + 27115460*x^12 + 5785190*x^11 - 52524876*x^10 - 37286821*x^9 + 46080424*x^8 + 57112611*x^7 - 6485584*x^6 - 31826729*x^5 - 10040832*x^4 + 4451908*x^3 + 3356864*x^2 + 718016*x + 51424)
 
gp: K = bnfinit(x^25 - 84*x^23 - 66*x^22 + 2826*x^21 + 4072*x^20 - 48221*x^19 - 96798*x^18 + 441736*x^17 + 1150294*x^16 - 2099287*x^15 - 7477592*x^14 + 3886133*x^13 + 27115460*x^12 + 5785190*x^11 - 52524876*x^10 - 37286821*x^9 + 46080424*x^8 + 57112611*x^7 - 6485584*x^6 - 31826729*x^5 - 10040832*x^4 + 4451908*x^3 + 3356864*x^2 + 718016*x + 51424, 1)
 

Normalized defining polynomial

\( x^{25} - 84 x^{23} - 66 x^{22} + 2826 x^{21} + 4072 x^{20} - 48221 x^{19} - 96798 x^{18} + 441736 x^{17} + 1150294 x^{16} - 2099287 x^{15} - 7477592 x^{14} + 3886133 x^{13} + 27115460 x^{12} + 5785190 x^{11} - 52524876 x^{10} - 37286821 x^{9} + 46080424 x^{8} + 57112611 x^{7} - 6485584 x^{6} - 31826729 x^{5} - 10040832 x^{4} + 4451908 x^{3} + 3356864 x^{2} + 718016 x + 51424 \)

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:  \(451947074858779414061989271610218361957576637330801=11^{20}\cdot 31^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.22$
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:  $11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(341=11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{341}(256,·)$, $\chi_{341}(1,·)$, $\chi_{341}(4,·)$, $\chi_{341}(70,·)$, $\chi_{341}(97,·)$, $\chi_{341}(64,·)$, $\chi_{341}(202,·)$, $\chi_{341}(78,·)$, $\chi_{341}(16,·)$, $\chi_{341}(280,·)$, $\chi_{341}(218,·)$, $\chi_{341}(157,·)$, $\chi_{341}(287,·)$, $\chi_{341}(225,·)$, $\chi_{341}(163,·)$, $\chi_{341}(295,·)$, $\chi_{341}(47,·)$, $\chi_{341}(221,·)$, $\chi_{341}(126,·)$, $\chi_{341}(311,·)$, $\chi_{341}(312,·)$, $\chi_{341}(159,·)$, $\chi_{341}(188,·)$, $\chi_{341}(125,·)$, $\chi_{341}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{20} + \frac{1}{10} a^{19} - \frac{1}{5} a^{18} + \frac{1}{10} a^{16} + \frac{1}{10} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{3}{10} a^{8} + \frac{2}{5} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{5} + \frac{2}{5} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{21} + \frac{1}{5} a^{19} + \frac{1}{5} a^{18} + \frac{1}{10} a^{17} - \frac{1}{10} a^{16} + \frac{1}{10} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{10} a^{11} + \frac{2}{5} a^{10} + \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{1340} a^{22} + \frac{3}{134} a^{21} - \frac{2}{67} a^{20} + \frac{12}{67} a^{19} - \frac{15}{67} a^{18} - \frac{143}{670} a^{17} - \frac{151}{1340} a^{16} - \frac{159}{670} a^{15} + \frac{14}{67} a^{14} + \frac{27}{670} a^{13} + \frac{279}{1340} a^{12} + \frac{77}{670} a^{11} + \frac{637}{1340} a^{10} + \frac{23}{134} a^{9} + \frac{199}{670} a^{8} - \frac{6}{67} a^{7} + \frac{317}{1340} a^{6} - \frac{149}{670} a^{5} - \frac{387}{1340} a^{4} + \frac{76}{335} a^{3} - \frac{11}{1340} a^{2} + \frac{13}{670} a + \frac{167}{335}$, $\frac{1}{1830440} a^{23} - \frac{111}{457610} a^{22} + \frac{529}{228805} a^{21} - \frac{7929}{183044} a^{20} - \frac{8723}{915220} a^{19} + \frac{6366}{228805} a^{18} - \frac{14109}{366088} a^{17} - \frac{17503}{915220} a^{16} + \frac{19178}{228805} a^{15} + \frac{64049}{915220} a^{14} - \frac{405743}{1830440} a^{13} - \frac{31013}{457610} a^{12} + \frac{453189}{1830440} a^{11} + \frac{23185}{91522} a^{10} - \frac{152131}{915220} a^{9} - \frac{37659}{91522} a^{8} - \frac{120497}{366088} a^{7} + \frac{15847}{45761} a^{6} - \frac{440829}{1830440} a^{5} - \frac{16019}{45761} a^{4} - \frac{162473}{366088} a^{3} + \frac{9645}{45761} a^{2} - \frac{204383}{457610} a - \frac{107718}{228805}$, $\frac{1}{512041959180649312867216537789099994176213958379975924056560} a^{24} - \frac{34200156688258790243025581378848826437400847031769359}{256020979590324656433608268894549997088106979189987962028280} a^{23} + \frac{27045211310502333421282135159383550454214392530295738793}{128010489795162328216804134447274998544053489594993981014140} a^{22} + \frac{1743783899154115664006967562176488780292975550945089542119}{256020979590324656433608268894549997088106979189987962028280} a^{21} + \frac{6615248542762704862704826737395527217582395848253664044463}{256020979590324656433608268894549997088106979189987962028280} a^{20} + \frac{31141888656354463676275829736277750784903978702295890171509}{128010489795162328216804134447274998544053489594993981014140} a^{19} - \frac{34510788754346284164826211621119290024363987727112901057069}{512041959180649312867216537789099994176213958379975924056560} a^{18} - \frac{4938554352209955246557248354981714724933994115521334087764}{32002622448790582054201033611818749636013372398748495253535} a^{17} - \frac{5720238501811088290657479723343644419095550860786072995481}{25602097959032465643360826889454999708810697918998796202828} a^{16} + \frac{33700049164267813763243299095131682699652973629394229953003}{256020979590324656433608268894549997088106979189987962028280} a^{15} - \frac{62543570539182685396234537371321440657122963388818138087451}{512041959180649312867216537789099994176213958379975924056560} a^{14} + \frac{45026801412667771363329580380464644613184744665012247409797}{256020979590324656433608268894549997088106979189987962028280} a^{13} - \frac{42213317619205292468085447305337758723626109994727498370667}{512041959180649312867216537789099994176213958379975924056560} a^{12} - \frac{56097342101600727085413320928206239926703379652138371664813}{256020979590324656433608268894549997088106979189987962028280} a^{11} + \frac{15629445515033277132752722549443665512547711376407922736119}{256020979590324656433608268894549997088106979189987962028280} a^{10} + \frac{8554163348377589071330784980174570038431203918005195412894}{32002622448790582054201033611818749636013372398748495253535} a^{9} - \frac{195405580595572236064988592038126539779022956249068522129301}{512041959180649312867216537789099994176213958379975924056560} a^{8} + \frac{125591601037477330368212219767888858249403143096788101191131}{256020979590324656433608268894549997088106979189987962028280} a^{7} - \frac{70084815759092568071889817256571016052788799165679634133957}{512041959180649312867216537789099994176213958379975924056560} a^{6} + \frac{69714520268596121358809119636034139114739063031727974684227}{256020979590324656433608268894549997088106979189987962028280} a^{5} - \frac{60867627249994536733047425225553243318832708706776475575161}{512041959180649312867216537789099994176213958379975924056560} a^{4} - \frac{49110951333679866396768513437616021158491707338064157840041}{256020979590324656433608268894549997088106979189987962028280} a^{3} + \frac{56518354906497096418415958719124523754013422937363595285803}{128010489795162328216804134447274998544053489594993981014140} a^{2} + \frac{2712866844408073293810554600915596659586187758659430969928}{32002622448790582054201033611818749636013372398748495253535} a - \frac{1107553271923041596537129759267486144594242737670901230396}{32002622448790582054201033611818749636013372398748495253535}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 238326232991215330 \) (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

5.5.13521270961.2, 5.5.13521270961.4, 5.5.13521270961.3, 5.5.13521270961.1, \(\Q(\zeta_{11})^+\), 5.5.923521.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
11Data not computed
31Data not computed