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Properties

Label 35.35.158...321.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.589\times 10^{95}$
Root discriminant \(524.84\)
Ramified prime $631$
Class number not computed
Class group not computed
Galois group $C_{35}$ (as 35T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879)
 
gp: K = bnfinit(y^35 - y^34 - 306*y^33 + 619*y^32 + 39918*y^31 - 119144*y^30 - 2870165*y^29 + 11295452*y^28 + 122756704*y^27 - 613544599*y^26 - 3125202817*y^25 + 20310067731*y^24 + 43327481459*y^23 - 419360814437*y^22 - 191553090102*y^21 + 5427052738052*y^20 - 3059836998013*y^19 - 43960753448720*y^18 + 54505011780373*y^17 + 222701516749019*y^16 - 391917990416865*y^15 - 698605421390367*y^14 + 1577201532402182*y^13 + 1315814631610304*y^12 - 3783939420646960*y^11 - 1378428158842980*y^10 + 5373712881752846*y^9 + 635624897449864*y^8 - 4223346678884758*y^7 + 43477310737712*y^6 + 1553138017862548*y^5 - 130695356406557*y^4 - 188318564137938*y^3 + 16596019359307*y^2 + 7161902265758*y - 508905734879, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^35 - x^34 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^35 - x^34 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879)
 

\( x^{35} - x^{34} - 306 x^{33} + 619 x^{32} + 39918 x^{31} - 119144 x^{30} - 2870165 x^{29} + \cdots - 508905734879 \) 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:   \(158\!\cdots\!321\) \(\medspace = 631^{34}\) 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:  \(524.84\)
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:  $631^{34/35}\approx 524.8419258377568$
Ramified primes:   \(631\) 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:  \(631\)
Dirichlet character group:    $\lbrace$$\chi_{631}(512,·)$, $\chi_{631}(1,·)$, $\chi_{631}(133,·)$, $\chi_{631}(269,·)$, $\chi_{631}(21,·)$, $\chi_{631}(279,·)$, $\chi_{631}(25,·)$, $\chi_{631}(5,·)$, $\chi_{631}(34,·)$, $\chi_{631}(36,·)$, $\chi_{631}(298,·)$, $\chi_{631}(427,·)$, $\chi_{631}(180,·)$, $\chi_{631}(182,·)$, $\chi_{631}(312,·)$, $\chi_{631}(441,·)$, $\chi_{631}(415,·)$, $\chi_{631}(579,·)$, $\chi_{631}(525,·)$, $\chi_{631}(464,·)$, $\chi_{631}(593,·)$, $\chi_{631}(83,·)$, $\chi_{631}(601,·)$, $\chi_{631}(219,·)$, $\chi_{631}(481,·)$, $\chi_{631}(228,·)$, $\chi_{631}(101,·)$, $\chi_{631}(105,·)$, $\chi_{631}(125,·)$, $\chi_{631}(625,·)$, $\chi_{631}(242,·)$, $\chi_{631}(371,·)$, $\chi_{631}(170,·)$, $\chi_{631}(505,·)$, $\chi_{631}(509,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{86}a^{23}+\frac{10}{43}a^{22}-\frac{11}{86}a^{21}+\frac{2}{43}a^{20}-\frac{7}{86}a^{19}-\frac{17}{86}a^{18}-\frac{4}{43}a^{17}+\frac{17}{86}a^{16}+\frac{10}{43}a^{15}+\frac{9}{43}a^{14}-\frac{13}{86}a^{13}-\frac{5}{43}a^{12}+\frac{2}{43}a^{11}-\frac{3}{86}a^{10}-\frac{18}{43}a^{9}+\frac{20}{43}a^{8}+\frac{33}{86}a^{7}-\frac{3}{43}a^{6}-\frac{19}{86}a^{5}+\frac{31}{86}a^{4}-\frac{19}{43}a^{3}+\frac{3}{86}a^{2}+\frac{4}{43}a-\frac{31}{86}$, $\frac{1}{86}a^{24}+\frac{19}{86}a^{22}+\frac{9}{86}a^{21}-\frac{1}{86}a^{20}+\frac{37}{86}a^{19}-\frac{6}{43}a^{18}-\frac{19}{43}a^{17}-\frac{19}{86}a^{16}+\frac{5}{86}a^{15}-\frac{29}{86}a^{14}+\frac{35}{86}a^{13}+\frac{16}{43}a^{12}+\frac{3}{86}a^{11}+\frac{12}{43}a^{10}-\frac{7}{43}a^{9}-\frac{18}{43}a^{8}+\frac{11}{43}a^{7}-\frac{14}{43}a^{6}+\frac{12}{43}a^{5}-\frac{13}{86}a^{4}-\frac{11}{86}a^{3}+\frac{17}{43}a^{2}-\frac{19}{86}a-\frac{25}{86}$, $\frac{1}{86}a^{25}+\frac{8}{43}a^{22}-\frac{7}{86}a^{21}+\frac{2}{43}a^{20}-\frac{4}{43}a^{19}+\frac{27}{86}a^{18}+\frac{2}{43}a^{17}+\frac{13}{43}a^{16}+\frac{21}{86}a^{15}-\frac{3}{43}a^{14}-\frac{11}{43}a^{13}-\frac{11}{43}a^{12}-\frac{9}{86}a^{11}-\frac{20}{43}a^{9}+\frac{18}{43}a^{8}-\frac{5}{43}a^{7}-\frac{17}{43}a^{6}-\frac{39}{86}a^{5}-\frac{41}{86}a^{4}+\frac{25}{86}a^{3}-\frac{33}{86}a^{2}-\frac{5}{86}a-\frac{13}{86}$, $\frac{1}{86}a^{26}+\frac{17}{86}a^{22}+\frac{4}{43}a^{21}+\frac{7}{43}a^{20}-\frac{33}{86}a^{19}+\frac{9}{43}a^{18}-\frac{9}{43}a^{17}+\frac{7}{86}a^{16}+\frac{9}{43}a^{15}+\frac{17}{43}a^{14}+\frac{7}{43}a^{13}-\frac{21}{86}a^{12}+\frac{11}{43}a^{11}+\frac{4}{43}a^{10}+\frac{5}{43}a^{9}+\frac{19}{43}a^{8}+\frac{20}{43}a^{7}-\frac{29}{86}a^{6}+\frac{5}{86}a^{5}-\frac{41}{86}a^{4}-\frac{27}{86}a^{3}+\frac{33}{86}a^{2}+\frac{31}{86}a-\frac{10}{43}$, $\frac{1}{86}a^{27}+\frac{6}{43}a^{22}-\frac{7}{43}a^{21}-\frac{15}{86}a^{20}-\frac{35}{86}a^{19}+\frac{13}{86}a^{18}+\frac{7}{43}a^{17}+\frac{15}{43}a^{16}-\frac{5}{86}a^{15}-\frac{17}{43}a^{14}-\frac{15}{86}a^{13}+\frac{10}{43}a^{12}+\frac{13}{43}a^{11}-\frac{25}{86}a^{10}-\frac{19}{43}a^{9}+\frac{5}{86}a^{8}+\frac{6}{43}a^{7}-\frac{11}{43}a^{6}-\frac{19}{86}a^{5}+\frac{5}{86}a^{4}-\frac{9}{86}a^{3}-\frac{10}{43}a^{2}+\frac{8}{43}a-\frac{16}{43}$, $\frac{1}{86}a^{28}+\frac{2}{43}a^{22}-\frac{6}{43}a^{21}+\frac{3}{86}a^{20}+\frac{11}{86}a^{19}-\frac{20}{43}a^{18}-\frac{3}{86}a^{17}+\frac{3}{43}a^{16}+\frac{27}{86}a^{15}+\frac{27}{86}a^{14}-\frac{39}{86}a^{13}-\frac{13}{43}a^{12}+\frac{13}{86}a^{11}-\frac{1}{43}a^{10}+\frac{7}{86}a^{9}+\frac{5}{86}a^{8}+\frac{6}{43}a^{7}+\frac{5}{43}a^{6}+\frac{9}{43}a^{5}+\frac{3}{43}a^{4}+\frac{3}{43}a^{3}-\frac{10}{43}a^{2}-\frac{21}{43}a-\frac{15}{86}$, $\frac{1}{86}a^{29}-\frac{3}{43}a^{22}+\frac{2}{43}a^{21}-\frac{5}{86}a^{20}-\frac{6}{43}a^{19}-\frac{21}{86}a^{18}-\frac{5}{86}a^{17}+\frac{1}{43}a^{16}-\frac{5}{43}a^{15}-\frac{25}{86}a^{14}-\frac{17}{86}a^{13}-\frac{33}{86}a^{12}-\frac{9}{43}a^{11}+\frac{19}{86}a^{10}-\frac{23}{86}a^{9}-\frac{19}{86}a^{8}-\frac{18}{43}a^{7}-\frac{1}{86}a^{6}+\frac{39}{86}a^{5}+\frac{11}{86}a^{4}-\frac{20}{43}a^{3}+\frac{16}{43}a^{2}+\frac{39}{86}a-\frac{5}{86}$, $\frac{1}{172}a^{30}-\frac{1}{172}a^{29}-\frac{1}{172}a^{28}-\frac{1}{172}a^{26}-\frac{1}{172}a^{25}-\frac{1}{172}a^{24}+\frac{31}{172}a^{22}-\frac{15}{86}a^{21}-\frac{3}{172}a^{20}+\frac{7}{43}a^{19}-\frac{9}{43}a^{18}+\frac{7}{86}a^{17}+\frac{35}{86}a^{16}-\frac{9}{172}a^{15}+\frac{47}{172}a^{14}-\frac{39}{172}a^{13}-\frac{2}{43}a^{12}-\frac{27}{86}a^{11}+\frac{41}{86}a^{10}-\frac{23}{86}a^{9}+\frac{51}{172}a^{8}-\frac{23}{86}a^{7}+\frac{85}{172}a^{6}-\frac{16}{43}a^{5}+\frac{9}{172}a^{4}+\frac{23}{172}a^{3}-\frac{75}{172}a^{2}-\frac{1}{43}a+\frac{21}{172}$, $\frac{1}{172}a^{31}-\frac{1}{172}a^{28}-\frac{1}{172}a^{27}-\frac{1}{172}a^{24}-\frac{1}{172}a^{23}+\frac{17}{172}a^{22}-\frac{15}{172}a^{21}+\frac{9}{172}a^{20}+\frac{6}{43}a^{19}+\frac{27}{86}a^{18}-\frac{21}{86}a^{17}+\frac{17}{172}a^{16}+\frac{29}{86}a^{15}+\frac{10}{43}a^{14}-\frac{25}{172}a^{13}+\frac{5}{43}a^{12}-\frac{6}{43}a^{11}-\frac{18}{43}a^{10}+\frac{19}{172}a^{9}-\frac{47}{172}a^{8}-\frac{83}{172}a^{7}-\frac{1}{172}a^{6}+\frac{47}{172}a^{5}-\frac{35}{86}a^{4}-\frac{19}{86}a^{3}-\frac{25}{172}a^{2}+\frac{63}{172}a-\frac{9}{172}$, $\frac{1}{172}a^{32}-\frac{1}{172}a^{29}-\frac{1}{172}a^{28}-\frac{1}{172}a^{25}-\frac{1}{172}a^{24}-\frac{1}{172}a^{23}-\frac{31}{172}a^{22}+\frac{35}{172}a^{21}+\frac{19}{86}a^{20}-\frac{39}{86}a^{19}+\frac{3}{86}a^{18}+\frac{75}{172}a^{17}-\frac{19}{43}a^{16}-\frac{31}{86}a^{15}-\frac{5}{172}a^{14}+\frac{41}{86}a^{13}+\frac{35}{86}a^{12}-\frac{29}{86}a^{11}-\frac{13}{172}a^{10}-\frac{1}{172}a^{9}-\frac{29}{172}a^{8}-\frac{79}{172}a^{7}-\frac{17}{172}a^{6}+\frac{7}{86}a^{5}-\frac{20}{43}a^{4}+\frac{57}{172}a^{3}-\frac{77}{172}a^{2}-\frac{67}{172}a-\frac{11}{43}$, $\frac{1}{172}a^{33}-\frac{1}{172}a^{28}-\frac{1}{172}a^{23}-\frac{4}{43}a^{22}-\frac{9}{43}a^{21}-\frac{23}{172}a^{20}+\frac{25}{86}a^{19}+\frac{69}{172}a^{18}+\frac{7}{86}a^{17}+\frac{17}{86}a^{16}-\frac{17}{86}a^{15}+\frac{15}{172}a^{14}-\frac{81}{172}a^{13}-\frac{6}{43}a^{12}-\frac{37}{172}a^{11}+\frac{7}{172}a^{10}-\frac{85}{172}a^{9}+\frac{3}{86}a^{8}+\frac{13}{172}a^{7}+\frac{79}{172}a^{6}-\frac{27}{86}a^{5}+\frac{27}{86}a^{4}-\frac{5}{86}a^{3}-\frac{3}{86}a^{2}-\frac{15}{43}a-\frac{3}{172}$, $\frac{1}{75\!\cdots\!28}a^{34}+\frac{24\!\cdots\!49}{37\!\cdots\!14}a^{33}-\frac{51\!\cdots\!40}{18\!\cdots\!07}a^{32}-\frac{79\!\cdots\!65}{37\!\cdots\!14}a^{31}+\frac{63\!\cdots\!03}{75\!\cdots\!28}a^{30}+\frac{10\!\cdots\!17}{37\!\cdots\!14}a^{29}-\frac{25\!\cdots\!23}{75\!\cdots\!28}a^{28}-\frac{53\!\cdots\!67}{37\!\cdots\!14}a^{27}+\frac{10\!\cdots\!37}{75\!\cdots\!28}a^{26}-\frac{32\!\cdots\!63}{75\!\cdots\!28}a^{25}-\frac{82\!\cdots\!31}{37\!\cdots\!14}a^{24}+\frac{34\!\cdots\!25}{37\!\cdots\!14}a^{23}-\frac{41\!\cdots\!95}{75\!\cdots\!28}a^{22}-\frac{12\!\cdots\!09}{75\!\cdots\!28}a^{21}+\frac{42\!\cdots\!97}{75\!\cdots\!28}a^{20}-\frac{34\!\cdots\!11}{75\!\cdots\!28}a^{19}-\frac{41\!\cdots\!61}{18\!\cdots\!07}a^{18}+\frac{12\!\cdots\!77}{37\!\cdots\!14}a^{17}+\frac{15\!\cdots\!41}{37\!\cdots\!14}a^{16}-\frac{80\!\cdots\!85}{37\!\cdots\!14}a^{15}+\frac{12\!\cdots\!89}{37\!\cdots\!14}a^{14}+\frac{82\!\cdots\!83}{75\!\cdots\!28}a^{13}-\frac{91\!\cdots\!29}{75\!\cdots\!28}a^{12}-\frac{29\!\cdots\!67}{75\!\cdots\!28}a^{11}+\frac{22\!\cdots\!17}{75\!\cdots\!28}a^{10}-\frac{46\!\cdots\!35}{37\!\cdots\!14}a^{9}+\frac{19\!\cdots\!04}{18\!\cdots\!07}a^{8}+\frac{44\!\cdots\!79}{17\!\cdots\!96}a^{7}-\frac{16\!\cdots\!63}{75\!\cdots\!28}a^{6}-\frac{87\!\cdots\!83}{18\!\cdots\!07}a^{5}-\frac{12\!\cdots\!47}{75\!\cdots\!28}a^{4}+\frac{24\!\cdots\!21}{75\!\cdots\!28}a^{3}-\frac{39\!\cdots\!45}{17\!\cdots\!96}a^{2}-\frac{16\!\cdots\!39}{75\!\cdots\!28}a+\frac{95\!\cdots\!41}{75\!\cdots\!28}$ 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:  $2$

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 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879)
 
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 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879, 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 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879);
 
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 - 306*x^33 + 619*x^32 + 39918*x^31 - 119144*x^30 - 2870165*x^29 + 11295452*x^28 + 122756704*x^27 - 613544599*x^26 - 3125202817*x^25 + 20310067731*x^24 + 43327481459*x^23 - 419360814437*x^22 - 191553090102*x^21 + 5427052738052*x^20 - 3059836998013*x^19 - 43960753448720*x^18 + 54505011780373*x^17 + 222701516749019*x^16 - 391917990416865*x^15 - 698605421390367*x^14 + 1577201532402182*x^13 + 1315814631610304*x^12 - 3783939420646960*x^11 - 1378428158842980*x^10 + 5373712881752846*x^9 + 635624897449864*x^8 - 4223346678884758*x^7 + 43477310737712*x^6 + 1553138017862548*x^5 - 130695356406557*x^4 - 188318564137938*x^3 + 16596019359307*x^2 + 7161902265758*x - 508905734879);
 
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 $C_{35}$

Intermediate fields

5.5.158532181921.1, 7.7.63121332085847281.1

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 ${\href{/padicField/2.5.0.1}{5} }^{7}$ $35$ $35$ $35$ ${\href{/padicField/11.5.0.1}{5} }^{7}$ $35$ $35$ $35$ $35$ $35$ $35$ ${\href{/padicField/37.7.0.1}{7} }^{5}$ ${\href{/padicField/41.7.0.1}{7} }^{5}$ ${\href{/padicField/43.1.0.1}{1} }^{35}$ ${\href{/padicField/47.5.0.1}{5} }^{7}$ $35$ $35$

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
\(631\) Copy content Toggle raw display Deg $35$$35$$1$$34$