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Properties

Label 35.35.168...681.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.681\times 10^{89}$
Root discriminant \(354.24\)
Ramified prime $421$
Class number not computed
Class group not computed
Galois group $C_{35}$ (as 35T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171)
 
gp: K = bnfinit(y^35 - y^34 - 204*y^33 + 437*y^32 + 17534*y^31 - 54290*y^30 - 824862*y^29 + 3275941*y^28 + 23317913*y^27 - 115242097*y^26 - 407103864*y^25 + 2577740505*y^24 + 4218284220*y^23 - 38412158946*y^22 - 19912715245*y^21 + 391393011501*y^20 - 73963916883*y^19 - 2766871309456*y^18 + 1816877075105*y^17 + 13678303310453*y^16 - 13286645514747*y^15 - 47446655533795*y^14 + 55834172665771*y^13 + 115251505419203*y^12 - 148585020568878*y^11 - 193921735357495*y^10 + 253198008831294*y^9 + 219700193289987*y^8 - 266521489745621*y^7 - 156623228737724*y^6 + 157601287591422*y^5 + 59099363823325*y^4 - 42232493776557*y^3 - 6358323449919*y^2 + 3052649440546*y - 157964821171, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171)
 

\( x^{35} - x^{34} - 204 x^{33} + 437 x^{32} + 17534 x^{31} - 54290 x^{30} - 824862 x^{29} + \cdots - 157964821171 \) Copy content Toggle raw display

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(168\!\cdots\!681\) \(\medspace = 421^{34}\) 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:  \(354.24\)
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:  $421^{34/35}\approx 354.2440850381781$
Ramified primes:   \(421\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $35$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(421\)
Dirichlet character group:    $\lbrace$$\chi_{421}(1,·)$, $\chi_{421}(385,·)$, $\chi_{421}(139,·)$, $\chi_{421}(279,·)$, $\chi_{421}(152,·)$, $\chi_{421}(27,·)$, $\chi_{421}(414,·)$, $\chi_{421}(33,·)$, $\chi_{421}(290,·)$, $\chi_{421}(291,·)$, $\chi_{421}(296,·)$, $\chi_{421}(48,·)$, $\chi_{421}(49,·)$, $\chi_{421}(307,·)$, $\chi_{421}(308,·)$, $\chi_{421}(315,·)$, $\chi_{421}(60,·)$, $\chi_{421}(317,·)$, $\chi_{421}(190,·)$, $\chi_{421}(321,·)$, $\chi_{421}(68,·)$, $\chi_{421}(199,·)$, $\chi_{421}(75,·)$, $\chi_{421}(78,·)$, $\chi_{421}(85,·)$, $\chi_{421}(354,·)$, $\chi_{421}(357,·)$, $\chi_{421}(232,·)$, $\chi_{421}(366,·)$, $\chi_{421}(370,·)$, $\chi_{421}(247,·)$, $\chi_{421}(376,·)$, $\chi_{421}(377,·)$, $\chi_{421}(252,·)$, $\chi_{421}(341,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{29}a^{21}-\frac{5}{29}a^{20}-\frac{10}{29}a^{19}+\frac{10}{29}a^{18}-\frac{11}{29}a^{17}-\frac{4}{29}a^{16}-\frac{9}{29}a^{15}-\frac{9}{29}a^{14}+\frac{9}{29}a^{13}-\frac{3}{29}a^{12}+\frac{12}{29}a^{11}+\frac{7}{29}a^{10}+\frac{2}{29}a^{9}+\frac{13}{29}a^{8}-\frac{14}{29}a^{7}-\frac{11}{29}a^{6}+\frac{3}{29}a^{5}+\frac{6}{29}a^{4}+\frac{14}{29}a^{3}+\frac{10}{29}a^{2}-\frac{11}{29}a$, $\frac{1}{29}a^{22}-\frac{6}{29}a^{20}-\frac{11}{29}a^{19}+\frac{10}{29}a^{18}-\frac{1}{29}a^{17}+\frac{4}{29}a^{15}-\frac{7}{29}a^{14}+\frac{13}{29}a^{13}-\frac{3}{29}a^{12}+\frac{9}{29}a^{11}+\frac{8}{29}a^{10}-\frac{6}{29}a^{9}-\frac{7}{29}a^{8}+\frac{6}{29}a^{7}+\frac{6}{29}a^{6}-\frac{8}{29}a^{5}-\frac{14}{29}a^{4}-\frac{7}{29}a^{3}+\frac{10}{29}a^{2}+\frac{3}{29}a$, $\frac{1}{29}a^{23}-\frac{12}{29}a^{20}+\frac{8}{29}a^{19}+\frac{1}{29}a^{18}-\frac{8}{29}a^{17}+\frac{9}{29}a^{16}-\frac{3}{29}a^{15}-\frac{12}{29}a^{14}-\frac{7}{29}a^{13}-\frac{9}{29}a^{12}-\frac{7}{29}a^{11}+\frac{7}{29}a^{10}+\frac{5}{29}a^{9}-\frac{3}{29}a^{8}+\frac{9}{29}a^{7}+\frac{13}{29}a^{6}+\frac{4}{29}a^{5}+\frac{7}{29}a^{3}+\frac{5}{29}a^{2}-\frac{8}{29}a$, $\frac{1}{29}a^{24}+\frac{6}{29}a^{20}-\frac{3}{29}a^{19}-\frac{4}{29}a^{18}-\frac{7}{29}a^{17}+\frac{7}{29}a^{16}-\frac{4}{29}a^{15}+\frac{1}{29}a^{14}+\frac{12}{29}a^{13}-\frac{14}{29}a^{12}+\frac{6}{29}a^{11}+\frac{2}{29}a^{10}-\frac{8}{29}a^{9}-\frac{9}{29}a^{8}-\frac{10}{29}a^{7}-\frac{12}{29}a^{6}+\frac{7}{29}a^{5}-\frac{8}{29}a^{4}-\frac{1}{29}a^{3}-\frac{4}{29}a^{2}+\frac{13}{29}a$, $\frac{1}{29}a^{25}-\frac{2}{29}a^{20}-\frac{2}{29}a^{19}-\frac{9}{29}a^{18}-\frac{14}{29}a^{17}-\frac{9}{29}a^{16}-\frac{3}{29}a^{15}+\frac{8}{29}a^{14}-\frac{10}{29}a^{13}-\frac{5}{29}a^{12}-\frac{12}{29}a^{11}+\frac{8}{29}a^{10}+\frac{8}{29}a^{9}-\frac{1}{29}a^{8}+\frac{14}{29}a^{7}-\frac{14}{29}a^{6}+\frac{3}{29}a^{5}-\frac{8}{29}a^{4}-\frac{1}{29}a^{3}+\frac{11}{29}a^{2}+\frac{8}{29}a$, $\frac{1}{29}a^{26}-\frac{12}{29}a^{20}+\frac{6}{29}a^{18}-\frac{2}{29}a^{17}-\frac{11}{29}a^{16}-\frac{10}{29}a^{15}+\frac{1}{29}a^{14}+\frac{13}{29}a^{13}+\frac{11}{29}a^{12}+\frac{3}{29}a^{11}-\frac{7}{29}a^{10}+\frac{3}{29}a^{9}+\frac{11}{29}a^{8}-\frac{13}{29}a^{7}+\frac{10}{29}a^{6}-\frac{2}{29}a^{5}+\frac{11}{29}a^{4}+\frac{10}{29}a^{3}-\frac{1}{29}a^{2}+\frac{7}{29}a$, $\frac{1}{29}a^{27}-\frac{2}{29}a^{20}+\frac{2}{29}a^{19}+\frac{2}{29}a^{18}+\frac{2}{29}a^{17}+\frac{9}{29}a^{15}-\frac{8}{29}a^{14}+\frac{3}{29}a^{13}-\frac{4}{29}a^{12}-\frac{8}{29}a^{11}+\frac{6}{29}a^{9}-\frac{2}{29}a^{8}-\frac{13}{29}a^{7}+\frac{11}{29}a^{6}-\frac{11}{29}a^{5}-\frac{5}{29}a^{4}-\frac{7}{29}a^{3}+\frac{11}{29}a^{2}+\frac{13}{29}a$, $\frac{1}{29}a^{28}-\frac{8}{29}a^{20}+\frac{11}{29}a^{19}-\frac{7}{29}a^{18}+\frac{7}{29}a^{17}+\frac{1}{29}a^{16}+\frac{3}{29}a^{15}+\frac{14}{29}a^{14}+\frac{14}{29}a^{13}-\frac{14}{29}a^{12}-\frac{5}{29}a^{11}-\frac{9}{29}a^{10}+\frac{2}{29}a^{9}+\frac{13}{29}a^{8}+\frac{12}{29}a^{7}-\frac{4}{29}a^{6}+\frac{1}{29}a^{5}+\frac{5}{29}a^{4}+\frac{10}{29}a^{3}+\frac{4}{29}a^{2}+\frac{7}{29}a$, $\frac{1}{29}a^{29}-\frac{1}{29}a$, $\frac{1}{29}a^{30}-\frac{1}{29}a^{2}$, $\frac{1}{6757}a^{31}+\frac{2}{233}a^{30}+\frac{12}{6757}a^{29}-\frac{31}{6757}a^{28}+\frac{21}{6757}a^{27}+\frac{59}{6757}a^{26}+\frac{18}{6757}a^{25}+\frac{26}{6757}a^{24}-\frac{65}{6757}a^{23}+\frac{9}{6757}a^{22}+\frac{26}{6757}a^{21}-\frac{1787}{6757}a^{20}-\frac{2713}{6757}a^{19}+\frac{487}{6757}a^{18}-\frac{3141}{6757}a^{17}+\frac{623}{6757}a^{16}+\frac{2651}{6757}a^{15}-\frac{934}{6757}a^{14}-\frac{116}{233}a^{13}+\frac{2127}{6757}a^{12}+\frac{2083}{6757}a^{11}-\frac{1183}{6757}a^{10}-\frac{1832}{6757}a^{9}+\frac{219}{6757}a^{8}+\frac{1571}{6757}a^{7}-\frac{32}{233}a^{6}-\frac{3095}{6757}a^{5}-\frac{2079}{6757}a^{4}+\frac{2834}{6757}a^{3}-\frac{3226}{6757}a^{2}+\frac{1316}{6757}a-\frac{73}{233}$, $\frac{1}{78577153}a^{32}+\frac{5723}{78577153}a^{31}-\frac{1225761}{78577153}a^{30}+\frac{726174}{78577153}a^{29}+\frac{1054646}{78577153}a^{28}+\frac{327559}{78577153}a^{27}+\frac{1316581}{78577153}a^{26}-\frac{1068596}{78577153}a^{25}+\frac{357158}{78577153}a^{24}+\frac{437731}{78577153}a^{23}+\frac{826435}{78577153}a^{22}+\frac{703072}{78577153}a^{21}+\frac{7230102}{78577153}a^{20}-\frac{29372890}{78577153}a^{19}+\frac{25435235}{78577153}a^{18}+\frac{26719877}{78577153}a^{17}-\frac{8583588}{78577153}a^{16}+\frac{15369510}{78577153}a^{15}-\frac{24198463}{78577153}a^{14}-\frac{11545809}{78577153}a^{13}-\frac{23999154}{78577153}a^{12}+\frac{2794028}{78577153}a^{11}-\frac{7422798}{78577153}a^{10}-\frac{25683831}{78577153}a^{9}-\frac{2673126}{78577153}a^{8}+\frac{22476396}{78577153}a^{7}-\frac{20118858}{78577153}a^{6}+\frac{35643501}{78577153}a^{5}+\frac{37274828}{78577153}a^{4}-\frac{27953695}{78577153}a^{3}+\frac{1636342}{78577153}a^{2}+\frac{7961662}{78577153}a-\frac{502318}{2709557}$, $\frac{1}{78577153}a^{33}+\frac{1448}{78577153}a^{31}-\frac{1007788}{78577153}a^{30}+\frac{237431}{78577153}a^{29}-\frac{588465}{78577153}a^{28}-\frac{34482}{78577153}a^{27}-\frac{839480}{78577153}a^{26}-\frac{259081}{78577153}a^{25}-\frac{408719}{78577153}a^{24}-\frac{1084425}{78577153}a^{23}-\frac{1167967}{78577153}a^{22}-\frac{724674}{78577153}a^{21}-\frac{17030740}{78577153}a^{20}+\frac{19835328}{78577153}a^{19}-\frac{26825250}{78577153}a^{18}-\frac{130398}{337241}a^{17}-\frac{4259142}{78577153}a^{16}-\frac{30090800}{78577153}a^{15}+\frac{9545618}{78577153}a^{14}-\frac{9473737}{78577153}a^{13}+\frac{30208318}{78577153}a^{12}+\frac{22418842}{78577153}a^{11}-\frac{19305411}{78577153}a^{10}+\frac{25764780}{78577153}a^{9}+\frac{15651901}{78577153}a^{8}-\frac{17524007}{78577153}a^{7}+\frac{6589452}{78577153}a^{6}-\frac{12126945}{78577153}a^{5}+\frac{30517527}{78577153}a^{4}-\frac{14509954}{78577153}a^{3}+\frac{30349405}{78577153}a^{2}-\frac{13469099}{78577153}a+\frac{1344675}{2709557}$, $\frac{1}{41\!\cdots\!81}a^{34}+\frac{80\!\cdots\!45}{41\!\cdots\!81}a^{33}+\frac{40\!\cdots\!39}{41\!\cdots\!81}a^{32}+\frac{25\!\cdots\!74}{41\!\cdots\!81}a^{31}-\frac{67\!\cdots\!60}{41\!\cdots\!81}a^{30}-\frac{44\!\cdots\!63}{41\!\cdots\!81}a^{29}-\frac{57\!\cdots\!31}{41\!\cdots\!81}a^{28}+\frac{66\!\cdots\!87}{41\!\cdots\!81}a^{27}+\frac{68\!\cdots\!56}{41\!\cdots\!81}a^{26}+\frac{12\!\cdots\!51}{41\!\cdots\!81}a^{25}-\frac{61\!\cdots\!68}{41\!\cdots\!81}a^{24}+\frac{25\!\cdots\!97}{41\!\cdots\!81}a^{23}+\frac{55\!\cdots\!69}{41\!\cdots\!81}a^{22}+\frac{30\!\cdots\!36}{41\!\cdots\!81}a^{21}-\frac{15\!\cdots\!13}{41\!\cdots\!81}a^{20}+\frac{10\!\cdots\!98}{41\!\cdots\!81}a^{19}+\frac{84\!\cdots\!03}{49\!\cdots\!41}a^{18}-\frac{20\!\cdots\!78}{41\!\cdots\!81}a^{17}+\frac{49\!\cdots\!87}{14\!\cdots\!89}a^{16}-\frac{21\!\cdots\!67}{41\!\cdots\!81}a^{15}-\frac{12\!\cdots\!54}{41\!\cdots\!81}a^{14}-\frac{19\!\cdots\!90}{41\!\cdots\!81}a^{13}+\frac{12\!\cdots\!11}{41\!\cdots\!81}a^{12}+\frac{56\!\cdots\!01}{41\!\cdots\!81}a^{11}-\frac{15\!\cdots\!81}{41\!\cdots\!81}a^{10}-\frac{69\!\cdots\!18}{41\!\cdots\!81}a^{9}+\frac{14\!\cdots\!19}{41\!\cdots\!81}a^{8}-\frac{15\!\cdots\!29}{41\!\cdots\!81}a^{7}-\frac{10\!\cdots\!53}{41\!\cdots\!81}a^{6}+\frac{11\!\cdots\!51}{41\!\cdots\!81}a^{5}+\frac{12\!\cdots\!08}{41\!\cdots\!81}a^{4}-\frac{93\!\cdots\!64}{41\!\cdots\!81}a^{3}+\frac{20\!\cdots\!44}{41\!\cdots\!81}a^{2}-\frac{83\!\cdots\!55}{41\!\cdots\!81}a+\frac{67\!\cdots\!78}{14\!\cdots\!89}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $29$

Class group and class number

not computed

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:  $34$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171)
 
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^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171, 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^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171);
 
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^35 - x^34 - 204*x^33 + 437*x^32 + 17534*x^31 - 54290*x^30 - 824862*x^29 + 3275941*x^28 + 23317913*x^27 - 115242097*x^26 - 407103864*x^25 + 2577740505*x^24 + 4218284220*x^23 - 38412158946*x^22 - 19912715245*x^21 + 391393011501*x^20 - 73963916883*x^19 - 2766871309456*x^18 + 1816877075105*x^17 + 13678303310453*x^16 - 13286645514747*x^15 - 47446655533795*x^14 + 55834172665771*x^13 + 115251505419203*x^12 - 148585020568878*x^11 - 193921735357495*x^10 + 253198008831294*x^9 + 219700193289987*x^8 - 266521489745621*x^7 - 156623228737724*x^6 + 157601287591422*x^5 + 59099363823325*x^4 - 42232493776557*x^3 - 6358323449919*x^2 + 3052649440546*x - 157964821171);
 
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

$C_{35}$ (as 35T1):

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 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

5.5.31414372081.1, 7.7.5567914722008521.1

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 $35$ $35$ $35$ $35$ $35$ ${\href{/padicField/13.5.0.1}{5} }^{7}$ $35$ $35$ $35$ ${\href{/padicField/29.1.0.1}{1} }^{35}$ $35$ $35$ $35$ $35$ $35$ $35$ $35$

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
\(421\) Copy content Toggle raw display Deg $35$$35$$1$$34$