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Properties

Label 30.0.59555410366...6875.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,5^{51}\cdot 7^{25}$
Root discriminant $78.07$
Ramified primes $5, 7$
Class number $8672$ (GRH)
Class group $[2, 2, 2, 2, 542]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1715449, 170720, 52840, -643315, 4952075, -1555327, 7291300, -2644945, 8210425, -4365750, 4269742, 3003635, -438230, 2820, -349200, -760873, -130600, 91800, 8040, 52200, 29608, 4400, 4710, -335, -925, -101, -120, -5, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 10*x^28 - 5*x^27 - 120*x^26 - 101*x^25 - 925*x^24 - 335*x^23 + 4710*x^22 + 4400*x^21 + 29608*x^20 + 52200*x^19 + 8040*x^18 + 91800*x^17 - 130600*x^16 - 760873*x^15 - 349200*x^14 + 2820*x^13 - 438230*x^12 + 3003635*x^11 + 4269742*x^10 - 4365750*x^9 + 8210425*x^8 - 2644945*x^7 + 7291300*x^6 - 1555327*x^5 + 4952075*x^4 - 643315*x^3 + 52840*x^2 + 170720*x + 1715449)
 
gp: K = bnfinit(x^30 + 10*x^28 - 5*x^27 - 120*x^26 - 101*x^25 - 925*x^24 - 335*x^23 + 4710*x^22 + 4400*x^21 + 29608*x^20 + 52200*x^19 + 8040*x^18 + 91800*x^17 - 130600*x^16 - 760873*x^15 - 349200*x^14 + 2820*x^13 - 438230*x^12 + 3003635*x^11 + 4269742*x^10 - 4365750*x^9 + 8210425*x^8 - 2644945*x^7 + 7291300*x^6 - 1555327*x^5 + 4952075*x^4 - 643315*x^3 + 52840*x^2 + 170720*x + 1715449, 1)
 

Normalized defining polynomial

\( x^{30} + 10 x^{28} - 5 x^{27} - 120 x^{26} - 101 x^{25} - 925 x^{24} - 335 x^{23} + 4710 x^{22} + 4400 x^{21} + 29608 x^{20} + 52200 x^{19} + 8040 x^{18} + 91800 x^{17} - 130600 x^{16} - 760873 x^{15} - 349200 x^{14} + 2820 x^{13} - 438230 x^{12} + 3003635 x^{11} + 4269742 x^{10} - 4365750 x^{9} + 8210425 x^{8} - 2644945 x^{7} + 7291300 x^{6} - 1555327 x^{5} + 4952075 x^{4} - 643315 x^{3} + 52840 x^{2} + 170720 x + 1715449 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-595554103661284317897450790724178659729659557342529296875=-\,5^{51}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.07$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(1,·)$, $\chi_{175}(11,·)$, $\chi_{175}(69,·)$, $\chi_{175}(129,·)$, $\chi_{175}(139,·)$, $\chi_{175}(141,·)$, $\chi_{175}(89,·)$, $\chi_{175}(16,·)$, $\chi_{175}(81,·)$, $\chi_{175}(19,·)$, $\chi_{175}(46,·)$, $\chi_{175}(86,·)$, $\chi_{175}(151,·)$, $\chi_{175}(24,·)$, $\chi_{175}(36,·)$, $\chi_{175}(156,·)$, $\chi_{175}(94,·)$, $\chi_{175}(159,·)$, $\chi_{175}(34,·)$, $\chi_{175}(164,·)$, $\chi_{175}(104,·)$, $\chi_{175}(106,·)$, $\chi_{175}(71,·)$, $\chi_{175}(174,·)$, $\chi_{175}(51,·)$, $\chi_{175}(116,·)$, $\chi_{175}(54,·)$, $\chi_{175}(121,·)$, $\chi_{175}(59,·)$, $\chi_{175}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7} a^{24} - \frac{3}{7} a^{23} + \frac{3}{7} a^{22} + \frac{1}{7} a^{18} + \frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{1}{7} a^{12} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{6} - \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{25} + \frac{1}{7} a^{23} + \frac{2}{7} a^{22} + \frac{1}{7} a^{19} + \frac{3}{7} a^{18} + \frac{1}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{26} - \frac{2}{7} a^{23} - \frac{3}{7} a^{22} + \frac{1}{7} a^{20} + \frac{3}{7} a^{19} - \frac{2}{7} a^{17} - \frac{2}{7} a^{16} - \frac{2}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{4} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{2149} a^{27} + \frac{72}{2149} a^{26} + \frac{141}{2149} a^{25} + \frac{46}{2149} a^{24} - \frac{430}{2149} a^{23} + \frac{93}{307} a^{22} - \frac{62}{2149} a^{21} - \frac{632}{2149} a^{20} - \frac{99}{307} a^{19} - \frac{10}{307} a^{18} - \frac{516}{2149} a^{17} + \frac{320}{2149} a^{16} + \frac{491}{2149} a^{15} - \frac{772}{2149} a^{14} + \frac{979}{2149} a^{13} + \frac{1020}{2149} a^{12} + \frac{575}{2149} a^{11} - \frac{244}{2149} a^{10} + \frac{207}{2149} a^{9} - \frac{933}{2149} a^{8} - \frac{757}{2149} a^{7} - \frac{58}{307} a^{6} + \frac{467}{2149} a^{5} - \frac{67}{2149} a^{4} + \frac{159}{2149} a^{3} + \frac{132}{2149} a^{2} + \frac{104}{2149} a + \frac{118}{307}$, $\frac{1}{19016501} a^{28} + \frac{2824}{19016501} a^{27} - \frac{1334873}{19016501} a^{26} + \frac{1338550}{19016501} a^{25} + \frac{1059442}{19016501} a^{24} - \frac{2424217}{19016501} a^{23} + \frac{2338871}{19016501} a^{22} - \frac{3637593}{19016501} a^{21} - \frac{1530276}{19016501} a^{20} - \frac{1158153}{2716643} a^{19} + \frac{6357303}{19016501} a^{18} - \frac{4346343}{19016501} a^{17} + \frac{7232654}{19016501} a^{16} - \frac{7311852}{19016501} a^{15} - \frac{1070998}{2716643} a^{14} + \frac{621443}{19016501} a^{13} - \frac{9165385}{19016501} a^{12} - \frac{7725470}{19016501} a^{11} - \frac{5761648}{19016501} a^{10} + \frac{8646515}{19016501} a^{9} + \frac{7931334}{19016501} a^{8} + \frac{9054597}{19016501} a^{7} - \frac{6210896}{19016501} a^{6} + \frac{3393286}{19016501} a^{5} - \frac{5681674}{19016501} a^{4} - \frac{6816403}{19016501} a^{3} + \frac{3699537}{19016501} a^{2} - \frac{1760042}{19016501} a + \frac{9065220}{19016501}$, $\frac{1}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{29} - \frac{918809451429968226413357132148406167163228539974252242178255113625140562308326398614117422376122214620743}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{28} - \frac{1156813888599730601650884393880869832486308235989191262214474533232241585524197140829616397143847596639386361}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{27} - \frac{1183419037282889471947508914630657872316174093904977516953148642449586630041412471082319103202192247638305416773}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{26} + \frac{1064856626701021396077100745577036615995383438598187346156322747455229755393149996048389237016328778985505938476}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{25} + \frac{338829193595307576983294074576514287934714224311364153119358851092737357689833453540559045454579727516842263879}{5157473212023450811448842782146138855632838965033563832354357027150880467105565523387917569317701419967391169399} a^{24} - \frac{6267149124866663129985514120189869288287846639989113164720856689657763141117814914182644354235458394593666955670}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{23} - \frac{17413896107535242695521293314356862972351407207312560861979920253845222287469513556313391261337608023072323614724}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{22} + \frac{2554434667302676071261247626430203346337291888656212912733931504619982518596256831700156080294903820663127740132}{5157473212023450811448842782146138855632838965033563832354357027150880467105565523387917569317701419967391169399} a^{21} - \frac{31856083443928649977545785428596758424887490951298151289426117037152830430946638892785711269093835118165242141}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{20} + \frac{8253187593076335797841054718258972649711355189473443107175031995941029363953817370638343987173770254676190109826}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{19} + \frac{12875377384847489337010603172098072143676734432930202907790933825734776574139391352599980184425569935981931363539}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{18} + \frac{1972012623453671926151573696697412842502527801135002084232694921129293932783976310733205964882499343559159001799}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{17} - \frac{16431497987326062077487218473941642938088454205791768717741191900228296993670213015405286468522057954445606694104}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{16} - \frac{12259791172514568136173962091891953696716551651379477312193897359743412563868582783086547383700787472063359393075}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{15} + \frac{14052561652895505414271600337272055516364469281307107939455777417224423939745351940875726748165139002760318881780}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{14} - \frac{7291823099296959971856645989055053466810274924868580076782550450396624260167203648585533256249762500576691294267}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{13} - \frac{153214310318676263698061834491184599736874118951915190006076052145314716263211798875563017969022712618337022489}{736781887431921544492691826020876979376119852147651976050622432450125781015080789055416795616814488566770167057} a^{12} + \frac{3144405273028505543129563340257393113320683342224413106384398184913433028487274293425072218996438336760000205230}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{11} + \frac{3667456303352491534326697386735038694671526872488092676156826832144006333453855828112566765075332564610243948150}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{10} - \frac{8734365554872367479984224850393569056472436573628350810376720365305790882976338186642539815363308229928373799276}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{9} + \frac{8710122957030347776884335459304011517512297693027521677853557816270974943985685598592037632512579978744661114163}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{8} + \frac{15869635007549214882809374599487900576480275306786093846774947655574744043917356652912455055984668844749575137512}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{7} + \frac{4112857552291786850924232254687590734581386503249179385770154158402861168465065077394242498992541189665396017295}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{6} - \frac{1409513910571722472361636699530801793512179932305649892951127810762632105555279684277697696173143646980359811338}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{5} - \frac{15852318040601875081864228186522905928069413671253485568994439660180273751035523603698591509256687126940993019811}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{4} + \frac{9949914936544804307811617310328211025328073295098532655286443542487113781873432717928118335827692707450557842736}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a^{3} + \frac{1613031567944046875567173556142713033897716711737546591582393944421135327221412481261028072271091867986155194344}{5157473212023450811448842782146138855632838965033563832354357027150880467105565523387917569317701419967391169399} a^{2} + \frac{8925453379368793739054980915881574500108383363614907877421994992068558534618692483843577830710078968856121502150}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793} a + \frac{5436053362861414118520763545183973593921242812755588956956070368960844769362269928001034717236994122608967809999}{36102312484164155680141899475022971989429872755234946826480499190056163269738958663715422985223909939771738185793}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{542}$, which has order $8672$ (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:  $14$
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:  \( 1429364684924.111 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), 5.5.390625.1, 6.0.2100875.1, 10.0.12822723388671875.1, 15.15.16836836874485015869140625.1

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 $30$ $15^{2}$ R R $15^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ $15^{2}$ $30$ $30$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $30$ $30$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $15^{2}$ $30$ $30$

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
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$