Properties

Label 30.0.112...787.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.128\times 10^{49}$
Root discriminant $43.16$
Ramified primes $3, 7, 11$
Class number $976$ (GRH)
Class group $[2, 2, 2, 122]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 23*x^28 - 12*x^27 + 335*x^26 - 144*x^25 + 2773*x^24 - 863*x^23 + 16295*x^22 - 4775*x^21 + 62257*x^20 - 15750*x^19 + 170334*x^18 - 52802*x^17 + 293956*x^16 - 111720*x^15 + 369164*x^14 - 135917*x^13 + 276687*x^12 - 78965*x^11 + 139349*x^10 - 31906*x^9 + 42847*x^8 - 4541*x^7 + 8009*x^6 - 477*x^5 + 1036*x^4 + 2*x^3 + 63*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^30 - x^29 + 23*x^28 - 12*x^27 + 335*x^26 - 144*x^25 + 2773*x^24 - 863*x^23 + 16295*x^22 - 4775*x^21 + 62257*x^20 - 15750*x^19 + 170334*x^18 - 52802*x^17 + 293956*x^16 - 111720*x^15 + 369164*x^14 - 135917*x^13 + 276687*x^12 - 78965*x^11 + 139349*x^10 - 31906*x^9 + 42847*x^8 - 4541*x^7 + 8009*x^6 - 477*x^5 + 1036*x^4 + 2*x^3 + 63*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 63, 2, 1036, -477, 8009, -4541, 42847, -31906, 139349, -78965, 276687, -135917, 369164, -111720, 293956, -52802, 170334, -15750, 62257, -4775, 16295, -863, 2773, -144, 335, -12, 23, -1, 1]);
 

\( x^{30} - x^{29} + 23 x^{28} - 12 x^{27} + 335 x^{26} - 144 x^{25} + 2773 x^{24} - 863 x^{23} + 16295 x^{22} - 4775 x^{21} + 62257 x^{20} - 15750 x^{19} + 170334 x^{18} - 52802 x^{17} + 293956 x^{16} - 111720 x^{15} + 369164 x^{14} - 135917 x^{13} + 276687 x^{12} - 78965 x^{11} + 139349 x^{10} - 31906 x^{9} + 42847 x^{8} - 4541 x^{7} + 8009 x^{6} - 477 x^{5} + 1036 x^{4} + 2 x^{3} + 63 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-11277272245002111679540357002131401262707502951787\)\(\medspace = -\,3^{15}\cdot 7^{20}\cdot 11^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $43.16$
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:  $3, 7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(130,·)$, $\chi_{231}(67,·)$, $\chi_{231}(4,·)$, $\chi_{231}(86,·)$, $\chi_{231}(71,·)$, $\chi_{231}(137,·)$, $\chi_{231}(16,·)$, $\chi_{231}(148,·)$, $\chi_{231}(214,·)$, $\chi_{231}(23,·)$, $\chi_{231}(25,·)$, $\chi_{231}(218,·)$, $\chi_{231}(155,·)$, $\chi_{231}(92,·)$, $\chi_{231}(221,·)$, $\chi_{231}(158,·)$, $\chi_{231}(163,·)$, $\chi_{231}(100,·)$, $\chi_{231}(37,·)$, $\chi_{231}(169,·)$, $\chi_{231}(170,·)$, $\chi_{231}(113,·)$, $\chi_{231}(179,·)$, $\chi_{231}(53,·)$, $\chi_{231}(212,·)$, $\chi_{231}(58,·)$, $\chi_{231}(190,·)$, $\chi_{231}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{559} a^{25} + \frac{153}{559} a^{24} - \frac{72}{559} a^{23} + \frac{17}{559} a^{22} + \frac{277}{559} a^{21} + \frac{64}{559} a^{20} + \frac{9}{43} a^{19} + \frac{94}{559} a^{18} + \frac{98}{559} a^{17} + \frac{223}{559} a^{16} - \frac{48}{559} a^{15} - \frac{37}{559} a^{14} + \frac{152}{559} a^{13} + \frac{171}{559} a^{12} + \frac{62}{559} a^{11} + \frac{45}{559} a^{10} - \frac{232}{559} a^{9} - \frac{106}{559} a^{8} + \frac{268}{559} a^{7} + \frac{12}{43} a^{6} - \frac{89}{559} a^{5} + \frac{60}{559} a^{4} - \frac{266}{559} a^{3} + \frac{97}{559} a^{2} + \frac{80}{559} a + \frac{271}{559}$, $\frac{1}{559} a^{26} - \frac{3}{559} a^{24} - \frac{147}{559} a^{23} - \frac{88}{559} a^{22} + \frac{167}{559} a^{21} - \frac{4}{13} a^{20} + \frac{81}{559} a^{19} + \frac{250}{559} a^{18} - \frac{237}{559} a^{17} - \frac{68}{559} a^{16} + \frac{40}{559} a^{15} + \frac{223}{559} a^{14} - \frac{166}{559} a^{13} + \frac{4}{13} a^{12} + \frac{62}{559} a^{11} + \frac{150}{559} a^{10} + \frac{173}{559} a^{9} + \frac{275}{559} a^{8} - \frac{41}{559} a^{7} + \frac{80}{559} a^{6} + \frac{261}{559} a^{5} + \frac{57}{559} a^{4} - \frac{12}{559} a^{3} - \frac{227}{559} a^{2} - \frac{230}{559} a - \frac{97}{559}$, $\frac{1}{7453147} a^{27} + \frac{5318}{7453147} a^{26} + \frac{5442}{7453147} a^{25} + \frac{3213976}{7453147} a^{24} + \frac{57579}{573319} a^{23} + \frac{1217898}{7453147} a^{22} + \frac{138915}{573319} a^{21} - \frac{755077}{7453147} a^{20} - \frac{283589}{7453147} a^{19} + \frac{3647222}{7453147} a^{18} + \frac{464964}{7453147} a^{17} + \frac{1535190}{7453147} a^{16} + \frac{54439}{7453147} a^{15} - \frac{1991833}{7453147} a^{14} + \frac{1397306}{7453147} a^{13} - \frac{2988938}{7453147} a^{12} - \frac{45963}{111241} a^{11} - \frac{3357550}{7453147} a^{10} - \frac{110407}{7453147} a^{9} - \frac{1166291}{7453147} a^{8} + \frac{1135651}{7453147} a^{7} - \frac{749577}{7453147} a^{6} + \frac{2754858}{7453147} a^{5} + \frac{2622090}{7453147} a^{4} + \frac{1567677}{7453147} a^{3} + \frac{1439359}{7453147} a^{2} - \frac{3222641}{7453147} a + \frac{3132582}{7453147}$, $\frac{1}{101725029597347} a^{28} - \frac{3442475}{101725029597347} a^{27} - \frac{6492269699}{7825002276719} a^{26} + \frac{66857423989}{101725029597347} a^{25} + \frac{20225800327570}{101725029597347} a^{24} + \frac{48813119521087}{101725029597347} a^{23} - \frac{48123554680025}{101725029597347} a^{22} + \frac{20943291860421}{101725029597347} a^{21} + \frac{27562979945096}{101725029597347} a^{20} + \frac{40894153588173}{101725029597347} a^{19} - \frac{2068338725831}{101725029597347} a^{18} - \frac{26633229480333}{101725029597347} a^{17} - \frac{28345662164034}{101725029597347} a^{16} - \frac{45727029003182}{101725029597347} a^{15} + \frac{39852430332899}{101725029597347} a^{14} + \frac{3194433112281}{101725029597347} a^{13} - \frac{38075735517816}{101725029597347} a^{12} - \frac{906449876591}{101725029597347} a^{11} + \frac{800203541648}{7825002276719} a^{10} - \frac{3633669352494}{7825002276719} a^{9} - \frac{38258676150313}{101725029597347} a^{8} + \frac{6458408394895}{101725029597347} a^{7} - \frac{2266883133410}{101725029597347} a^{6} + \frac{44212067064012}{101725029597347} a^{5} - \frac{1698837855881}{7825002276719} a^{4} - \frac{7400841530825}{101725029597347} a^{3} - \frac{1534795111637}{101725029597347} a^{2} + \frac{28386429424084}{101725029597347} a - \frac{21101841351411}{101725029597347}$, $\frac{1}{3084773403942098639179250903059492060622687333} a^{29} + \frac{297600540368358256421219380406}{3084773403942098639179250903059492060622687333} a^{28} + \frac{3575219649609121915914618378920526470}{237290261841699895321480838696884004663283641} a^{27} - \frac{1741144586932700978370876090256414941473759}{3084773403942098639179250903059492060622687333} a^{26} - \frac{83012025437687660003430334291802368175272}{237290261841699895321480838696884004663283641} a^{25} + \frac{540835216848055091408264587598642912522632343}{3084773403942098639179250903059492060622687333} a^{24} - \frac{282745187163878112066191569316277377910709413}{3084773403942098639179250903059492060622687333} a^{23} - \frac{117215275429236555187136305895917969071236136}{3084773403942098639179250903059492060622687333} a^{22} + \frac{1379648409514634624318864438397415381143450695}{3084773403942098639179250903059492060622687333} a^{21} - \frac{801772493637437159563789333257956633156934373}{3084773403942098639179250903059492060622687333} a^{20} - \frac{656530043710668582276984670906928499295576505}{3084773403942098639179250903059492060622687333} a^{19} + \frac{100862835341980205564013068013763494577729939}{237290261841699895321480838696884004663283641} a^{18} - \frac{797059694748655316868699872702737300867145368}{3084773403942098639179250903059492060622687333} a^{17} + \frac{159740669673542463961167580993754230982785032}{3084773403942098639179250903059492060622687333} a^{16} + \frac{877976300696288174065174227421279848626093248}{3084773403942098639179250903059492060622687333} a^{15} - \frac{49700257131068118088075387740694320199993755}{237290261841699895321480838696884004663283641} a^{14} + \frac{782849247357294197132207176236604911689850790}{3084773403942098639179250903059492060622687333} a^{13} + \frac{1468317043864315448902662452822390789236017554}{3084773403942098639179250903059492060622687333} a^{12} - \frac{484955469743131098952334797482230754713612022}{3084773403942098639179250903059492060622687333} a^{11} + \frac{506736267188918416862330094527512422220914720}{3084773403942098639179250903059492060622687333} a^{10} + \frac{267167935266633028610790478767112825006401160}{3084773403942098639179250903059492060622687333} a^{9} + \frac{97541672865886148918509007457873999788487747}{3084773403942098639179250903059492060622687333} a^{8} - \frac{915245552615326085396222188957197646079969210}{3084773403942098639179250903059492060622687333} a^{7} + \frac{891781495725874667927871373017071502914794797}{3084773403942098639179250903059492060622687333} a^{6} - \frac{803859501930001016647424744586354158018695056}{3084773403942098639179250903059492060622687333} a^{5} + \frac{910662033814184654023162537829710448786930136}{3084773403942098639179250903059492060622687333} a^{4} + \frac{393462544117703292081221578974013591860457638}{3084773403942098639179250903059492060622687333} a^{3} - \frac{1384697963709719239914129026168008407575384560}{3084773403942098639179250903059492060622687333} a^{2} - \frac{354369320432158898881310775132513329637097410}{3084773403942098639179250903059492060622687333} a - \frac{995038117952492601888297240639645196618287919}{3084773403942098639179250903059492060622687333}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{122}$, which has order $976$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{33818387380679807589465361410172326167}{226013889917515988574891368211254183533} a^{29} + \frac{31338667935399949868997219900410037083}{226013889917515988574891368211254183533} a^{28} - \frac{59533394026528196135127009037326757884}{17385683839808922198068566785481091041} a^{27} + \frac{26720697277131697071025705847393655581}{17385683839808922198068566785481091041} a^{26} - \frac{11267162530505485654312665849674405751741}{226013889917515988574891368211254183533} a^{25} + \frac{4022194077956355207850219157836948393930}{226013889917515988574891368211254183533} a^{24} - \frac{92953188386234510257152391692858763228843}{226013889917515988574891368211254183533} a^{23} + \frac{22104963375560112489949892735920235290783}{226013889917515988574891368211254183533} a^{22} - \frac{545080356364611035579293293564068809395600}{226013889917515988574891368211254183533} a^{21} + \frac{119856710703093578498163583079545405844510}{226013889917515988574891368211254183533} a^{20} - \frac{159315389114482945879813536122692152315726}{17385683839808922198068566785481091041} a^{19} + \frac{371505998836784739401662931229174563050057}{226013889917515988574891368211254183533} a^{18} - \frac{5636380533746118790557453282431658208023094}{226013889917515988574891368211254183533} a^{17} + \frac{1341202572841470363726285604899046711111310}{226013889917515988574891368211254183533} a^{16} - \frac{9580462806832944693507608665472531054975619}{226013889917515988574891368211254183533} a^{15} + \frac{2975338497658454058333296233214938733630477}{226013889917515988574891368211254183533} a^{14} - \frac{274902467923716523011393173737372315661302}{5256136974825953222671892283982655431} a^{13} + \frac{3528124328723702882659130579676347333166157}{226013889917515988574891368211254183533} a^{12} - \frac{8545078952599196519129045655575977701620936}{226013889917515988574891368211254183533} a^{11} + \frac{1804537778175184053576987723405631500682837}{226013889917515988574891368211254183533} a^{10} - \frac{4181221389403621251758861434697384536802496}{226013889917515988574891368211254183533} a^{9} + \frac{636454856620926484319260785746287491702239}{226013889917515988574891368211254183533} a^{8} - \frac{1210070166745826883897941813261487755642144}{226013889917515988574891368211254183533} a^{7} + \frac{8202236392955944036524565106868987880557}{226013889917515988574891368211254183533} a^{6} - \frac{216060402937800866443945700291944166376263}{226013889917515988574891368211254183533} a^{5} - \frac{8171735372298450139059825122527343449906}{226013889917515988574891368211254183533} a^{4} - \frac{26753434861254060031931004678000206681808}{226013889917515988574891368211254183533} a^{3} - \frac{3975069744616359074607077694949359502761}{226013889917515988574891368211254183533} a^{2} - \frac{1348278749125616613229589043671236379186}{226013889917515988574891368211254183533} a + \frac{1714530982849230131954290236414673446}{3373341640559940127983453256884390799} \) (order $6$)
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:  \( 4697581952.048968 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 976}{6\sqrt{11277272245002111679540357002131401262707502951787}}\approx 0.213682484973285$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.64827.1, 10.0.52089208083.1, 15.15.886528337182930278529.1

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 $30$ R $30$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ $30$ $30$ $30$

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
3Data not computed
$7$7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
7.15.10.1$x^{15} + 4116 x^{6} - 2401 x^{3} + 1075648$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
11Data not computed