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Properties

Label 30.0.104...371.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.046\times 10^{45}$
Root discriminant \(31.67\)
Ramified primes $7,11$
Class number $16$ (GRH)
Class group [2, 2, 2, 2] (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^30 - y^29 + 3*y^28 - 4*y^27 + 9*y^26 - 14*y^25 + 28*y^24 - 47*y^23 + 89*y^22 - 155*y^21 + 286*y^20 + 132*y^19 + 285*y^18 + 265*y^17 + 437*y^16 + 378*y^15 + 761*y^14 + 432*y^13 + 1468*y^12 + 157*y^11 + 3211*y^10 - 1429*y^9 + 636*y^8 - 283*y^7 + 126*y^6 - 56*y^5 + 25*y^4 - 11*y^3 + 5*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
 

\( x^{30} - x^{29} + 3 x^{28} - 4 x^{27} + 9 x^{26} - 14 x^{25} + 28 x^{24} - 47 x^{23} + 89 x^{22} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1046076147688308987260717152173116396995512371\) \(\medspace = -\,7^{20}\cdot 11^{27}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(31.67\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{2/3}11^{9/10}\approx 31.670295092032053$
Ramified primes:   \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:  $30$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(2,·)$, $\chi_{77}(67,·)$, $\chi_{77}(4,·)$, $\chi_{77}(65,·)$, $\chi_{77}(8,·)$, $\chi_{77}(9,·)$, $\chi_{77}(74,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(18,·)$, $\chi_{77}(23,·)$, $\chi_{77}(25,·)$, $\chi_{77}(71,·)$, $\chi_{77}(29,·)$, $\chi_{77}(30,·)$, $\chi_{77}(32,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(39,·)$, $\chi_{77}(43,·)$, $\chi_{77}(46,·)$, $\chi_{77}(72,·)$, $\chi_{77}(50,·)$, $\chi_{77}(51,·)$, $\chi_{77}(53,·)$, $\chi_{77}(57,·)$, $\chi_{77}(58,·)$, $\chi_{77}(60,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{16384}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{696851}a^{21}-\frac{332692}{696851}a^{20}+\frac{335130}{696851}a^{19}-\frac{303662}{696851}a^{18}-\frac{55621}{696851}a^{17}-\frac{216573}{696851}a^{16}-\frac{198331}{696851}a^{15}-\frac{290436}{696851}a^{14}-\frac{322799}{696851}a^{13}+\frac{240447}{696851}a^{12}+\frac{217221}{696851}a^{11}+\frac{319182}{696851}a^{10}+\frac{341691}{696851}a^{9}+\frac{138309}{696851}a^{8}+\frac{167404}{696851}a^{7}-\frac{245946}{696851}a^{6}+\frac{22212}{696851}a^{5}-\frac{346700}{696851}a^{4}+\frac{145178}{696851}a^{3}-\frac{119515}{696851}a^{2}+\frac{63171}{696851}a-\frac{157023}{696851}$, $\frac{1}{696851}a^{22}-\frac{318543}{696851}a^{11}-\frac{163850}{696851}$, $\frac{1}{696851}a^{23}-\frac{318543}{696851}a^{12}-\frac{163850}{696851}a$, $\frac{1}{696851}a^{24}-\frac{318543}{696851}a^{13}-\frac{163850}{696851}a^{2}$, $\frac{1}{696851}a^{25}-\frac{318543}{696851}a^{14}-\frac{163850}{696851}a^{3}$, $\frac{1}{696851}a^{26}-\frac{318543}{696851}a^{15}-\frac{163850}{696851}a^{4}$, $\frac{1}{696851}a^{27}-\frac{318543}{696851}a^{16}-\frac{163850}{696851}a^{5}$, $\frac{1}{696851}a^{28}-\frac{318543}{696851}a^{17}-\frac{163850}{696851}a^{6}$, $\frac{1}{696851}a^{29}-\frac{318543}{696851}a^{18}-\frac{163850}{696851}a^{7}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{310114}{696851} a^{29} + \frac{310114}{696851} a^{28} - \frac{930342}{696851} a^{27} + \frac{1240456}{696851} a^{26} - \frac{2791026}{696851} a^{25} + \frac{4341596}{696851} a^{24} - \frac{8683192}{696851} a^{23} + \frac{14575358}{696851} a^{22} - \frac{27600301}{696851} a^{21} + \frac{48067670}{696851} a^{20} - \frac{88692604}{696851} a^{19} - \frac{40935048}{696851} a^{18} - \frac{88382490}{696851} a^{17} - \frac{82180210}{696851} a^{16} - \frac{135519818}{696851} a^{15} - \frac{117223092}{696851} a^{14} - \frac{235996754}{696851} a^{13} - \frac{133969248}{696851} a^{12} - \frac{455247352}{696851} a^{11} - \frac{48790154}{696851} a^{10} - \frac{995776054}{696851} a^{9} + \frac{443152906}{696851} a^{8} - \frac{197232504}{696851} a^{7} + \frac{87762262}{696851} a^{6} - \frac{39074364}{696851} a^{5} + \frac{17366384}{696851} a^{4} - \frac{7752850}{696851} a^{3} + \frac{3411254}{696851} a^{2} - \frac{1550570}{696851} a + \frac{620228}{696851} \)  (order $22$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{417}{696851}a^{23}+\frac{266110}{696851}a^{12}-\frac{2821456}{696851}a$, $\frac{344166}{696851}a^{29}-\frac{172083}{696851}a^{28}+\frac{860415}{696851}a^{27}-\frac{860415}{696851}a^{26}+\frac{2409162}{696851}a^{25}-\frac{3269577}{696851}a^{24}+\frac{7227486}{696851}a^{23}-\frac{11357478}{696851}a^{22}+\frac{22542873}{696851}a^{21}-\frac{38030057}{696851}a^{20}+\frac{71758611}{696851}a^{19}+\frac{94645650}{696851}a^{18}+\frac{120802266}{696851}a^{17}+\frac{140247645}{696851}a^{16}+\frac{196002537}{696851}a^{15}+\frac{205295019}{696851}a^{14}+\frac{326957700}{696851}a^{13}+\frac{279634875}{696851}a^{12}+\frac{579575544}{696851}a^{11}+\frac{306651906}{696851}a^{10}+\frac{1132318240}{696851}a^{9}+\frac{60745299}{696851}a^{8}-\frac{27017031}{696851}a^{7}+\frac{12045810}{696851}a^{6}-\frac{5334573}{696851}a^{5}+\frac{2409162}{696851}a^{4}-\frac{1032498}{696851}a^{3}+\frac{516249}{696851}a^{2}-\frac{172083}{696851}a+\frac{172083}{696851}$, $\frac{248706}{696851}a^{29}-\frac{138031}{696851}a^{28}+\frac{684710}{696851}a^{27}-\frac{690155}{696851}a^{26}+\frac{1931314}{696851}a^{25}-\frac{2622589}{696851}a^{24}+\frac{5797302}{696851}a^{23}-\frac{9110046}{696851}a^{22}+\frac{18082061}{696851}a^{21}-\frac{30504982}{696851}a^{20}+\frac{57558927}{696851}a^{19}+\frac{58436328}{696851}a^{18}+\frac{96897762}{696851}a^{17}+\frac{109015506}{696851}a^{16}+\frac{157217309}{696851}a^{15}+\frac{163954580}{696851}a^{14}+\frac{262258900}{696851}a^{13}+\frac{224300375}{696851}a^{12}+\frac{464888408}{696851}a^{11}+\frac{245971242}{696851}a^{10}+\frac{908024022}{696851}a^{9}+\frac{48724943}{696851}a^{8}+\frac{179854040}{696851}a^{7}+\frac{9662170}{696851}a^{6}+\frac{35635516}{696851}a^{5}+\frac{1932434}{696851}a^{4}+\frac{7077362}{696851}a^{3}+\frac{414093}{696851}a^{2}-\frac{138031}{696851}a-\frac{558820}{696851}$, $\frac{2371}{696851}a^{26}+\frac{417}{696851}a^{24}+\frac{1514733}{696851}a^{15}+\frac{266110}{696851}a^{13}-\frac{17763618}{696851}a^{4}-\frac{3518307}{696851}a^{2}$, $\frac{12141}{696851}a^{28}-\frac{286}{696851}a^{23}+\frac{7757848}{696851}a^{17}-\frac{184183}{696851}a^{12}-\frac{89687024}{696851}a^{6}+\frac{1565785}{696851}a$, $\frac{417}{696851}a^{24}-\frac{286}{696851}a^{23}+\frac{266110}{696851}a^{13}-\frac{184183}{696851}a^{12}-\frac{3518307}{696851}a^{2}+\frac{1565785}{696851}a-1$, $\frac{12141}{696851}a^{29}-\frac{172083}{696851}a^{28}+\frac{245632}{696851}a^{27}-\frac{552672}{696851}a^{26}+\frac{859712}{696851}a^{25}-\frac{1719424}{696851}a^{24}+\frac{2886176}{696851}a^{23}-\frac{5465312}{696851}a^{22}+\frac{9518240}{696851}a^{21}-\frac{17562688}{696851}a^{20}+\frac{31133677}{696851}a^{19}-\frac{48982965}{696851}a^{18}-\frac{8515272}{696851}a^{17}-\frac{26835296}{696851}a^{16}-\frac{23212224}{696851}a^{15}-\frac{46731488}{696851}a^{14}-\frac{26528256}{696851}a^{13}-\frac{90146944}{696851}a^{12}-\frac{9641056}{696851}a^{11}-\frac{197181088}{696851}a^{10}+\frac{87752032}{696851}a^{9}-\frac{491877849}{696851}a^{8}+\frac{380513801}{696851}a^{7}-\frac{97424432}{696851}a^{6}+\frac{3438848}{696851}a^{5}-\frac{1535200}{696851}a^{4}+\frac{675488}{696851}a^{3}-\frac{307040}{696851}a^{2}+\frac{122816}{696851}a-\frac{61408}{696851}$, $\frac{138031}{696851}a^{29}+\frac{276062}{696851}a^{27}-\frac{138031}{696851}a^{26}+\frac{690155}{696851}a^{25}-\frac{690155}{696851}a^{24}+\frac{1932434}{696851}a^{23}-\frac{2622589}{696851}a^{22}+\frac{5797433}{696851}a^{21}-\frac{9110177}{696851}a^{20}+\frac{18082061}{696851}a^{19}+\frac{57696958}{696851}a^{18}+\frac{57558927}{696851}a^{17}+\frac{75917050}{696851}a^{16}+\frac{96897762}{696851}a^{15}+\frac{112495265}{696851}a^{14}+\frac{157217309}{696851}a^{13}+\frac{164670983}{696851}a^{12}+\frac{262258900}{696851}a^{11}+\frac{224382302}{696851}a^{10}+\frac{464806481}{696851}a^{9}+\frac{245971242}{696851}a^{8}-\frac{109458583}{696851}a^{7}+\frac{48724943}{696851}a^{6}-\frac{21670867}{696851}a^{5}+\frac{9662170}{696851}a^{4}-\frac{4278961}{696851}a^{3}+\frac{1932434}{696851}a^{2}-\frac{828186}{696851}a-\frac{282758}{696851}$, $\frac{34052}{696851}a^{29}-\frac{184224}{696851}a^{28}+\frac{245632}{696851}a^{27}-\frac{552672}{696851}a^{26}+\frac{859712}{696851}a^{25}-\frac{1719138}{696851}a^{24}+\frac{2886176}{696851}a^{23}-\frac{5465312}{696851}a^{22}+\frac{9518240}{696851}a^{21}-\frac{17562688}{696851}a^{20}+\frac{31133677}{696851}a^{19}-\frac{34982002}{696851}a^{18}-\frac{16273120}{696851}a^{17}-\frac{26835296}{696851}a^{16}-\frac{23212224}{696851}a^{15}-\frac{46731488}{696851}a^{14}-\frac{26344073}{696851}a^{13}-\frac{90146944}{696851}a^{12}-\frac{9641056}{696851}a^{11}-\frac{197181088}{696851}a^{10}+\frac{87752032}{696851}a^{9}-\frac{491877849}{696851}a^{8}+\frac{218903371}{696851}a^{7}-\frac{7737408}{696851}a^{6}+\frac{3438848}{696851}a^{5}-\frac{1535200}{696851}a^{4}+\frac{675488}{696851}a^{3}-\frac{1872825}{696851}a^{2}+\frac{122816}{696851}a-\frac{61408}{696851}$, $\frac{12141}{696851}a^{28}-\frac{5445}{696851}a^{27}-\frac{286}{696851}a^{22}+\frac{7757848}{696851}a^{17}-\frac{3479759}{696851}a^{16}-\frac{184183}{696851}a^{11}-\frac{89687024}{696851}a^{6}+\frac{39914477}{696851}a^{5}+\frac{868934}{696851}$, $\frac{470056}{696851}a^{29}-\frac{248706}{696851}a^{28}+\frac{1243530}{696851}a^{27}-\frac{1243530}{696851}a^{26}+\frac{3481884}{696851}a^{25}-\frac{4725414}{696851}a^{24}+\frac{10445652}{696851}a^{23}-\frac{16414596}{696851}a^{22}+\frac{32580486}{696851}a^{21}-\frac{54964050}{696851}a^{20}+\frac{103710402}{696851}a^{19}+\frac{119307578}{696851}a^{18}+\frac{174591612}{696851}a^{17}+\frac{202695390}{696851}a^{16}+\frac{283276134}{696851}a^{15}+\frac{296706258}{696851}a^{14}+\frac{472541400}{696851}a^{13}+\frac{404147250}{696851}a^{12}+\frac{837641808}{696851}a^{11}+\frac{443194092}{696851}a^{10}+\frac{1636216445}{696851}a^{9}+\frac{87793218}{696851}a^{8}+\frac{162478065}{696851}a^{7}+\frac{17409420}{696851}a^{6}-\frac{7709886}{696851}a^{5}+\frac{3481884}{696851}a^{4}-\frac{1492236}{696851}a^{3}+\frac{746118}{696851}a^{2}-\frac{248706}{696851}a-\frac{448145}{696851}$, $\frac{122816}{696851}a^{29}-\frac{49267}{696851}a^{28}+\frac{307040}{696851}a^{27}-\frac{307040}{696851}a^{26}+\frac{859712}{696851}a^{25}-\frac{1166752}{696851}a^{24}+\frac{2578850}{696851}a^{23}-\frac{4052928}{696851}a^{22}+\frac{8044448}{696851}a^{21}-\frac{13570989}{696851}a^{20}+\frac{25607136}{696851}a^{19}+\frac{33774400}{696851}a^{18}+\frac{50866264}{696851}a^{17}+\frac{50047520}{696851}a^{16}+\frac{69943712}{696851}a^{15}+\frac{73259744}{696851}a^{14}+\frac{116675200}{696851}a^{13}+\frac{99603817}{696851}a^{12}+\frac{206822144}{696851}a^{11}+\frac{109429056}{696851}a^{10}+\frac{404125817}{696851}a^{9}+\frac{21677024}{696851}a^{8}-\frac{9641056}{696851}a^{7}-\frac{85388464}{696851}a^{6}-\frac{1903648}{696851}a^{5}+\frac{859712}{696851}a^{4}-\frac{368448}{696851}a^{3}+\frac{184224}{696851}a^{2}+\frac{1504377}{696851}a+\frac{61408}{696851}$, $\frac{620228}{696851}a^{29}-\frac{282758}{696851}a^{28}+\frac{1550570}{696851}a^{27}-\frac{1550570}{696851}a^{26}+\frac{4341596}{696851}a^{25}-\frac{5893000}{696851}a^{24}+\frac{13024788}{696851}a^{23}-\frac{20467524}{696851}a^{22}+\frac{40624934}{696851}a^{21}-\frac{68535039}{696851}a^{20}+\frac{129317538}{696851}a^{19}+\frac{170562700}{696851}a^{18}+\frac{235180750}{696851}a^{17}+\frac{252742910}{696851}a^{16}+\frac{353219846}{696851}a^{15}+\frac{369966002}{696851}a^{14}+\frac{588684380}{696851}a^{13}+\frac{503935250}{696851}a^{12}+\frac{1044463952}{696851}a^{11}+\frac{552623148}{696851}a^{10}+\frac{2040342262}{696851}a^{9}+\frac{109470242}{696851}a^{8}-\frac{48687898}{696851}a^{7}-\frac{179816927}{696851}a^{6}-\frac{9613534}{696851}a^{5}+\frac{4341596}{696851}a^{4}-\frac{1860684}{696851}a^{3}+\frac{7270105}{696851}a^{2}-\frac{310114}{696851}a+\frac{310114}{696851}$, $\frac{386737}{696851}a^{29}-\frac{552124}{696851}a^{28}+\frac{1236834}{696851}a^{27}-\frac{1930063}{696851}a^{26}+\frac{3863748}{696851}a^{25}-\frac{6487040}{696851}a^{24}+\frac{12284473}{696851}a^{23}-\frac{21394805}{696851}a^{22}+\frac{39476866}{696851}a^{21}-\frac{69981848}{696851}a^{20}+\frac{127540775}{696851}a^{19}+\frac{19097493}{696851}a^{18}+\frac{60319547}{696851}a^{17}+\frac{48695959}{696851}a^{16}+\frac{106556324}{696851}a^{15}+\frac{58912989}{696851}a^{14}+\frac{202895618}{696851}a^{13}+\frac{21486684}{696851}a^{12}+\frac{443217541}{696851}a^{11}-\frac{197246299}{696851}a^{10}+\frac{1105270321}{696851}a^{9}-\frac{1056545378}{696851}a^{8}+\frac{218916813}{696851}a^{7}-\frac{7729736}{696851}a^{6}+\frac{43365252}{696851}a^{5}-\frac{19281959}{696851}a^{4}+\frac{8595703}{696851}a^{3}-\frac{3794369}{696851}a^{2}+\frac{1703816}{696851}a-1$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4697581952.048968 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 16}{22\cdot\sqrt{1046076147688308987260717152173116396995512371}}\cr\approx \mathstrut & 0.0991946028104771 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.3195731.1, \(\Q(\zeta_{11})\), 15.15.886528337182930278529.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ $15^{2}$ $15^{2}$ R R ${\href{/padicField/13.10.0.1}{10} }^{3}$ $30$ $30$ ${\href{/padicField/23.3.0.1}{3} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{3}$ $15^{2}$ $15^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{15}$ $15^{2}$ $15^{2}$ $15^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display Deg $30$$3$$10$$20$
\(11\) Copy content Toggle raw display Deg $30$$10$$3$$27$