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Properties

Label 30.0.249...171.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.492\times 10^{50}$
Root discriminant \(47.85\)
Ramified primes $11,13$
Class number $427$ (GRH)
Class group [427] (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 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1)
 
gp: K = bnfinit(y^30 - y^29 + 5*y^28 - 10*y^27 + 31*y^26 - 76*y^25 + 210*y^24 - 545*y^23 + 1461*y^22 - 3851*y^21 + 10240*y^20 + 18326*y^19 + 26485*y^18 + 36579*y^17 + 51035*y^16 + 68796*y^15 + 98765*y^14 + 125384*y^13 + 200880*y^12 + 201891*y^11 + 476245*y^10 + 130439*y^9 + 35726*y^8 + 9785*y^7 + 2680*y^6 + 734*y^5 + 201*y^4 + 55*y^3 + 15*y^2 + 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1)
 

\( x^{30} - x^{29} + 5 x^{28} - 10 x^{27} + 31 x^{26} - 76 x^{25} + 210 x^{24} - 545 x^{23} + 1461 x^{22} + \cdots + 1 \) 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:   \(-249154964698353870876083085574129912384252301255171\) \(\medspace = -\,11^{27}\cdot 13^{20}\) 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:  \(47.85\)
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:  $11^{9/10}13^{2/3}\approx 47.8500414346374$
Ramified primes:   \(11\), \(13\) 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:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(68,·)$, $\chi_{143}(133,·)$, $\chi_{143}(9,·)$, $\chi_{143}(74,·)$, $\chi_{143}(139,·)$, $\chi_{143}(14,·)$, $\chi_{143}(79,·)$, $\chi_{143}(16,·)$, $\chi_{143}(81,·)$, $\chi_{143}(131,·)$, $\chi_{143}(87,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(29,·)$, $\chi_{143}(94,·)$, $\chi_{143}(35,·)$, $\chi_{143}(100,·)$, $\chi_{143}(40,·)$, $\chi_{143}(105,·)$, $\chi_{143}(42,·)$, $\chi_{143}(107,·)$, $\chi_{143}(48,·)$, $\chi_{143}(113,·)$, $\chi_{143}(53,·)$, $\chi_{143}(118,·)$, $\chi_{143}(120,·)$, $\chi_{143}(61,·)$, $\chi_{143}(126,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{16384}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5379069389}a^{21}-\frac{1152358415}{5379069389}a^{20}-\frac{2103974700}{5379069389}a^{19}-\frac{2505458961}{5379069389}a^{18}+\frac{620987965}{5379069389}a^{17}+\frac{2219289669}{5379069389}a^{16}-\frac{2608948237}{5379069389}a^{15}+\frac{106980170}{5379069389}a^{14}-\frac{2003924009}{5379069389}a^{13}-\frac{338276463}{5379069389}a^{12}-\frac{2405330354}{5379069389}a^{11}+\frac{1528096971}{5379069389}a^{10}+\frac{53078293}{5379069389}a^{9}-\frac{2458425821}{5379069389}a^{8}+\frac{1142642022}{5379069389}a^{7}-\frac{271284821}{5379069389}a^{6}+\frac{1921209341}{5379069389}a^{5}+\frac{1230078742}{5379069389}a^{4}+\frac{1346974054}{5379069389}a^{3}+\frac{1652131573}{5379069389}a^{2}+\frac{2505685901}{5379069389}a-\frac{2623203052}{5379069389}$, $\frac{1}{5379069389}a^{22}-\frac{1528051540}{5379069389}a^{11}-\frac{235676305}{5379069389}$, $\frac{1}{5379069389}a^{23}-\frac{1528051540}{5379069389}a^{12}-\frac{235676305}{5379069389}a$, $\frac{1}{5379069389}a^{24}-\frac{1528051540}{5379069389}a^{13}-\frac{235676305}{5379069389}a^{2}$, $\frac{1}{5379069389}a^{25}-\frac{1528051540}{5379069389}a^{14}-\frac{235676305}{5379069389}a^{3}$, $\frac{1}{5379069389}a^{26}-\frac{1528051540}{5379069389}a^{15}-\frac{235676305}{5379069389}a^{4}$, $\frac{1}{5379069389}a^{27}-\frac{1528051540}{5379069389}a^{16}-\frac{235676305}{5379069389}a^{5}$, $\frac{1}{5379069389}a^{28}-\frac{1528051540}{5379069389}a^{17}-\frac{235676305}{5379069389}a^{6}$, $\frac{1}{5379069389}a^{29}-\frac{1528051540}{5379069389}a^{18}-\frac{235676305}{5379069389}a^{7}$ 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

$C_{427}$, which has order $427$ (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:   \( \frac{44811}{5379069389} a^{24} + \frac{2035763030}{5379069389} a^{13} - \frac{71705594805}{5379069389} a^{2} \)  (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:   $\frac{44811}{5379069389}a^{24}+\frac{10240}{5379069389}a^{23}+\frac{2035763030}{5379069389}a^{13}+\frac{465083001}{5379069389}a^{12}-\frac{71705594805}{5379069389}a^{2}-\frac{19639485095}{5379069389}a$, $\frac{1473276186}{5379069389}a^{29}-\frac{1473276186}{5379069389}a^{28}+\frac{7368650975}{5379069389}a^{27}-\frac{14732761860}{5379069389}a^{26}+\frac{45671561766}{5379069389}a^{25}-\frac{111968990136}{5379069389}a^{24}+\frac{309387999060}{5379069389}a^{23}-\frac{802935521370}{5379069389}a^{22}+\frac{2152456503895}{5379069389}a^{21}-\frac{5673586592286}{5379069389}a^{20}+\frac{15086348144640}{5379069389}a^{19}+\frac{26999259384636}{5379069389}a^{18}+\frac{39019719786210}{5379069389}a^{17}+\frac{53994099997325}{5379069389}a^{16}+\frac{75188650152510}{5379069389}a^{15}+\frac{101355508492056}{5379069389}a^{14}+\frac{145508122510290}{5379069389}a^{13}+\frac{184725261305424}{5379069389}a^{12}+\frac{295951720243680}{5379069389}a^{11}+\frac{297441027036700}{5379069389}a^{10}+\frac{701640417201570}{5379069389}a^{9}+\frac{192172672425654}{5379069389}a^{8}+\frac{52634265021036}{5379069389}a^{7}+\frac{14416007480010}{5379069389}a^{6}+\frac{458408721845}{5379069389}a^{5}+\frac{1081384720524}{5379069389}a^{4}+\frac{296128513386}{5379069389}a^{3}+\frac{81030190230}{5379069389}a^{2}+\frac{22099142790}{5379069389}a+\frac{5893104744}{5379069389}$, $\frac{169004}{5379069389}a^{25}+\frac{44811}{5379069389}a^{24}+\frac{10240}{5379069389}a^{23}+\frac{7677969119}{5379069389}a^{14}+\frac{2035763030}{5379069389}a^{13}+\frac{465083001}{5379069389}a^{12}-\frac{261803824736}{5379069389}a^{3}-\frac{71705594805}{5379069389}a^{2}-\frac{19639485095}{5379069389}a$, $\frac{44811}{5379069389}a^{24}+\frac{2035763030}{5379069389}a^{13}-\frac{71705594805}{5379069389}a^{2}-1$, $\frac{555724556}{5379069389}a^{29}-\frac{555724556}{5379069389}a^{28}+\frac{2778622780}{5379069389}a^{27}-\frac{5557245560}{5379069389}a^{26}+\frac{17227461236}{5379069389}a^{25}-\frac{42235066256}{5379069389}a^{24}+\frac{116702156760}{5379069389}a^{23}-\frac{302869883020}{5379069389}a^{22}+\frac{811913594258}{5379069389}a^{21}-\frac{2140095265156}{5379069389}a^{20}+\frac{5690619453440}{5379069389}a^{19}+\frac{10184208213256}{5379069389}a^{18}+\frac{14718364865660}{5379069389}a^{17}+\frac{20327848533924}{5379069389}a^{16}+\frac{28361402715460}{5379069389}a^{15}+\frac{38231626554576}{5379069389}a^{14}+\frac{54886135773340}{5379069389}a^{13}+\frac{69678967729504}{5379069389}a^{12}+\frac{111633948809280}{5379069389}a^{11}+\frac{112196602280449}{5379069389}a^{10}+\frac{264661041172220}{5379069389}a^{9}+\frac{72488155360084}{5379069389}a^{8}+\frac{19853815487656}{5379069389}a^{7}+\frac{5437764780460}{5379069389}a^{6}+\frac{1489341810080}{5379069389}a^{5}+\frac{407901824104}{5379069389}a^{4}+\frac{111700635756}{5379069389}a^{3}+\frac{30564850580}{5379069389}a^{2}+\frac{8335868340}{5379069389}a+\frac{2222898224}{5379069389}$, $\frac{44811}{5379069389}a^{25}+\frac{2035763030}{5379069389}a^{14}-\frac{71705594805}{5379069389}a^{3}$, $\frac{8290211}{5379069389}a^{29}-\frac{145949969}{5379069389}a^{28}+\frac{300428295}{5379069389}a^{27}-\frac{937743889}{5379069389}a^{26}+\frac{2300338380}{5379069389}a^{25}-\frac{6356630569}{5379069389}a^{24}+\frac{16497049290}{5379069389}a^{23}-\frac{44224231325}{5379069389}a^{22}+\frac{116569130734}{5379069389}a^{21}-\frac{309963100160}{5379069389}a^{20}+\frac{820463837880}{5379069389}a^{19}-\frac{1800252810294}{5379069389}a^{18}-\frac{861949119002}{5379069389}a^{17}-\frac{1647951367821}{5379069389}a^{16}-\frac{2054232469419}{5379069389}a^{15}-\frac{2997278124129}{5379069389}a^{14}-\frac{3793782186227}{5379069389}a^{13}-\frac{6081069336921}{5379069389}a^{12}-\frac{6111382487120}{5379069389}a^{11}-\frac{14415857093330}{5379069389}a^{10}-\frac{3948366877126}{5379069389}a^{9}-\frac{47604425544845}{5379069389}a^{8}+\frac{33484603346911}{5379069389}a^{7}-\frac{8377493262201}{5379069389}a^{6}+\frac{3467753398479}{5379069389}a^{5}-\frac{961954455678}{5379069389}a^{4}+\frac{260138983866}{5379069389}a^{3}-\frac{47141087831}{5379069389}a^{2}+\frac{14139336370}{5379069389}a-\frac{1503546020}{5379069389}$, $\frac{456624843}{5379069389}a^{29}-\frac{611962280}{5379069389}a^{28}+\frac{2435094350}{5379069389}a^{27}-\frac{5326907016}{5379069389}a^{26}+\frac{15677497754}{5379069389}a^{25}-\frac{39421955019}{5379069389}a^{24}+\frac{107459046386}{5379069389}a^{23}-\frac{280824292536}{5379069389}a^{22}+\frac{750082384437}{5379069389}a^{21}-\frac{1980838593265}{5379069389}a^{20}+\frac{5261992482451}{5379069389}a^{19}+\frac{6809494075378}{5379069389}a^{18}+\frac{9162179259996}{5379069389}a^{17}+\frac{12660824480635}{5379069389}a^{16}+\frac{17753613853074}{5379069389}a^{15}+\frac{23648048841223}{5379069389}a^{14}+\frac{34626766636218}{5379069389}a^{13}+\frac{42220598441747}{5379069389}a^{12}+\frac{72641674842909}{5379069389}a^{11}+\frac{61611741093805}{5379069389}a^{10}+\frac{186734710698136}{5379069389}a^{9}-\frac{12926744888768}{5379069389}a^{8}-\frac{3540516824341}{5379069389}a^{7}+\frac{3836679692195}{5379069389}a^{6}+\frac{94953524422}{5379069389}a^{5}-\frac{668068802595}{5379069389}a^{4}-\frac{91644879616}{5379069389}a^{3}+\frac{14159986979}{5379069389}a^{2}-\frac{6901152199}{5379069389}a-\frac{3806693490}{5379069389}$, $\frac{1514965387}{5379069389}a^{29}-\frac{1684851347}{5379069389}a^{28}+\frac{7783272940}{5379069389}a^{27}-\frac{16024888945}{5379069389}a^{26}+\frac{48839941042}{5379069389}a^{25}-\frac{120723677535}{5379069389}a^{24}+\frac{332108627696}{5379069389}a^{23}-\frac{863843429940}{5379069389}a^{22}+\frac{2313001616946}{5379069389}a^{21}-\frac{6100484010526}{5379069389}a^{20}+\frac{16216333971091}{5379069389}a^{19}+\frac{25895120896151}{5379069389}a^{18}+\frac{37352609753578}{5379069389}a^{17}+\frac{51763361234659}{5379069389}a^{16}+\frac{72331419209691}{5379069389}a^{15}+\frac{97238074555291}{5379069389}a^{14}+\frac{140282999495136}{5379069389}a^{13}+\frac{176351375122571}{5379069389}a^{12}+\frac{287535686278616}{5379069389}a^{11}+\frac{277586753506455}{5379069389}a^{10}+\frac{696202519512331}{5379069389}a^{9}+\frac{126611668142161}{5379069389}a^{8}+\frac{52226336189251}{5379069389}a^{7}+\frac{19110679071776}{5379069389}a^{6}+\frac{3917780304946}{5379069389}a^{5}+\frac{712448216356}{5379069389}a^{4}+\frac{293835607331}{5379069389}a^{3}+\frac{8699257410}{5379069389}a^{2}+\frac{14524315855}{5379069389}a+\frac{3822414801}{5379069389}$, $\frac{2363687138}{5379069389}a^{29}-\frac{3342637547}{5379069389}a^{28}+\frac{12775406476}{5379069389}a^{27}-\frac{28433075535}{5379069389}a^{26}+\frac{82866540584}{5379069389}a^{25}-\frac{209376777871}{5379069389}a^{24}+\frac{569276591896}{5379069389}a^{23}-\frac{1489650064720}{5379069389}a^{22}+\frac{3976133165364}{5379069389}a^{21}-\frac{10504010034082}{5379069389}a^{20}+\frac{27898192733301}{5379069389}a^{19}+\frac{33494304381548}{5379069389}a^{18}+\frac{44127780513851}{5379069389}a^{17}+\frac{60952688022236}{5379069389}a^{16}+\frac{85542702730681}{5379069389}a^{15}+\frac{113649686566644}{5379069389}a^{14}+\frac{167453556449034}{5379069389}a^{13}+\frac{201628662354276}{5379069389}a^{12}+\frac{354544019927336}{5379069389}a^{11}+\frac{284515037277704}{5379069389}a^{10}+\frac{932031564132311}{5379069389}a^{9}-\frac{148516662966013}{5379069389}a^{8}-\frac{40677328062621}{5379069389}a^{7}+\frac{19149662932990}{5379069389}a^{6}+\frac{438481455140}{5379069389}a^{5}-\frac{835819067788}{5379069389}a^{4}+\frac{32822169520}{5379069389}a^{3}+\frac{80606575169}{5379069389}a^{2}+\frac{2336546460}{5379069389}a-\frac{4933863913}{5379069389}$, $\frac{4339579312}{5379069389}a^{29}-\frac{5923374578}{5379069389}a^{28}+\frac{23571798011}{5379069389}a^{27}-\frac{51566936555}{5379069389}a^{26}+\frac{151746995197}{5379069389}a^{25}-\frac{381578532174}{5379069389}a^{24}+\frac{1040132942505}{5379069389}a^{23}-\frac{2718194563170}{5379069389}a^{22}+\frac{7260305048459}{5379069389}a^{21}-\frac{19173216280753}{5379069389}a^{20}+\frac{50932631026206}{5379069389}a^{19}+\frac{62265626510567}{5379069389}a^{18}+\frac{88684711158230}{5379069389}a^{17}+\frac{122624978006427}{5379069389}a^{16}+\frac{171571850460205}{5379069389}a^{15}+\frac{228857342262332}{5379069389}a^{14}+\frac{335168859038814}{5379069389}a^{13}+\frac{408665625642952}{5379069389}a^{12}+\frac{703129899425616}{5379069389}a^{11}+\frac{596372062590524}{5379069389}a^{10}+\frac{18\!\cdots\!10}{5379069389}a^{9}-\frac{125122399924608}{5379069389}a^{8}+\frac{89066955764142}{5379069389}a^{7}+\frac{37136762874355}{5379069389}a^{6}-\frac{1614996725526}{5379069389}a^{5}+\frac{2785745439727}{5379069389}a^{4}+\frac{501067213942}{5379069389}a^{3}-\frac{53037942696}{5379069389}a^{2}+\frac{56972832945}{5379069389}a+\frac{959240831}{5379069389}$, $\frac{2070689943}{5379069389}a^{29}-\frac{2020710531}{5379069389}a^{28}+\frac{10150402911}{5379069389}a^{27}-\frac{20288527344}{5379069389}a^{26}+\frac{62899192006}{5379069389}a^{25}-\frac{154203987250}{5379069389}a^{24}+\frac{426090155820}{5379069389}a^{23}-\frac{1105805404390}{5379069389}a^{22}+\frac{2964370098153}{5379069389}a^{21}-\frac{7813681857442}{5379069389}a^{20}+\frac{20776967598080}{5379069389}a^{19}+\frac{39077449721869}{5379069389}a^{18}+\frac{54114717210830}{5379069389}a^{17}+\frac{74464109280502}{5379069389}a^{16}+\frac{103617294258037}{5379069389}a^{15}+\frac{139594813015751}{5379069389}a^{14}+\frac{200397399643688}{5379069389}a^{13}+\frac{254404229034928}{5379069389}a^{12}+\frac{407585669052960}{5379069389}a^{11}+\frac{409637629317149}{5379069389}a^{10}+\frac{966301458373790}{5379069389}a^{9}+\frac{264660827785738}{5379069389}a^{8}+\frac{8416464620125}{5379069389}a^{7}+\frac{7111560477710}{5379069389}a^{6}-\frac{2858648118521}{5379069389}a^{5}-\frac{783010868227}{5379069389}a^{4}+\frac{146025324406}{5379069389}a^{3}+\frac{12841890779}{5379069389}a^{2}+\frac{30435011130}{5379069389}a+\frac{8116002968}{5379069389}$, $\frac{525454722}{5379069389}a^{29}-\frac{564014767}{5379069389}a^{28}+\frac{2776352735}{5379069389}a^{27}-\frac{5557866525}{5379069389}a^{26}+\frac{17227292232}{5379069389}a^{25}-\frac{42235111067}{5379069389}a^{24}+\frac{116702156760}{5379069389}a^{23}-\frac{302869883020}{5379069389}a^{22}+\frac{811913594258}{5379069389}a^{21}-\frac{2140095265156}{5379069389}a^{20}+\frac{5690619453440}{5379069389}a^{19}+\frac{8809019397492}{5379069389}a^{18}+\frac{14341732306700}{5379069389}a^{17}+\frac{20224718144293}{5379069389}a^{16}+\frac{28333191685015}{5379069389}a^{15}+\frac{38223948585457}{5379069389}a^{14}+\frac{54884100010310}{5379069389}a^{13}+\frac{69678967729504}{5379069389}a^{12}+\frac{111633948809280}{5379069389}a^{11}+\frac{112196602280449}{5379069389}a^{10}+\frac{264661041172220}{5379069389}a^{9}+\frac{72488155360084}{5379069389}a^{8}+\frac{66376820943017}{5379069389}a^{7}+\frac{18179976563220}{5379069389}a^{6}+\frac{4979313266715}{5379069389}a^{5}+\frac{1363772043148}{5379069389}a^{4}+\frac{373504460492}{5379069389}a^{3}+\frac{102270445385}{5379069389}a^{2}+\frac{8335868340}{5379069389}a+\frac{2222898224}{5379069389}$, $\frac{8290211}{5379069389}a^{27}+\frac{2270045}{5379069389}a^{26}+\frac{169004}{5379069389}a^{25}+\frac{169004}{5379069389}a^{24}+\frac{44811}{5379069389}a^{23}+\frac{376632558960}{5379069389}a^{16}+\frac{103130389631}{5379069389}a^{15}+\frac{7677969119}{5379069389}a^{14}+\frac{7677969119}{5379069389}a^{13}+\frac{2035763030}{5379069389}a^{12}-\frac{12742211782760}{5379069389}a^{5}-\frac{3489971456635}{5379069389}a^{4}-\frac{261803824736}{5379069389}a^{3}-\frac{261803824736}{5379069389}a^{2}-\frac{71705594805}{5379069389}a$ 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:  \( 85915831770.81862 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 85915831770.81862 \cdot 427}{22\cdot\sqrt{249154964698353870876083085574129912384252301255171}}\cr\approx \mathstrut & 0.0992068467156376 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*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 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*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 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*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 + 5*x^28 - 10*x^27 + 31*x^26 - 76*x^25 + 210*x^24 - 545*x^23 + 1461*x^22 - 3851*x^21 + 10240*x^20 + 18326*x^19 + 26485*x^18 + 36579*x^17 + 51035*x^16 + 68796*x^15 + 98765*x^14 + 125384*x^13 + 200880*x^12 + 201891*x^11 + 476245*x^10 + 130439*x^9 + 35726*x^8 + 9785*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*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

\(\Q(\sqrt{-11}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 6.0.38014691.1, \(\Q(\zeta_{11})\), 15.15.432659002790862279847129.1

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 $30$ $15^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{6}$ $30$ R R $30$ $30$ ${\href{/padicField/23.3.0.1}{3} }^{10}$ $30$ ${\href{/padicField/31.5.0.1}{5} }^{6}$ $15^{2}$ $30$ ${\href{/padicField/43.6.0.1}{6} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{6}$ ${\href{/padicField/53.5.0.1}{5} }^{6}$ $15^{2}$

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
\(11\) Copy content Toggle raw display Deg $30$$10$$3$$27$
\(13\) Copy content Toggle raw display Deg $30$$3$$10$$20$