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Properties

Label 34.34.342...149.1
Degree $34$
Signature $[34, 0]$
Discriminant $3.425\times 10^{74}$
Root discriminant \(155.67\)
Ramified primes $3,103$
Class number not computed
Class group not computed
Galois group $C_{34}$ (as 34T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967)
 
gp: K = bnfinit(y^34 - y^33 - 101*y^32 - 39*y^31 + 4475*y^30 + 7317*y^29 - 107886*y^28 - 308922*y^27 + 1428035*y^26 + 6473545*y^25 - 8038724*y^24 - 76649821*y^23 - 34999669*y^22 + 511384499*y^21 + 872682200*y^20 - 1610027003*y^19 - 5837837426*y^18 - 597495179*y^17 + 18112177824*y^16 + 20594193249*y^15 - 20555590855*y^14 - 57678181587*y^13 - 18084925848*y^12 + 57474448334*y^11 + 58215655753*y^10 - 9898232087*y^9 - 42024711035*y^8 - 14514531077*y^7 + 10323419848*y^6 + 7567081746*y^5 - 122533277*y^4 - 1123034742*y^3 - 160716846*y^2 + 51273995*y + 10497967, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967)
 

\( x^{34} - x^{33} - 101 x^{32} - 39 x^{31} + 4475 x^{30} + 7317 x^{29} - 107886 x^{28} - 308922 x^{27} + \cdots + 10497967 \) Copy content Toggle raw display

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

Invariants

Degree:  $34$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[34, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(342523005011894297428856269332610453116457630461733441736562419892654124149\) \(\medspace = 3^{17}\cdot 103^{33}\) 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:  \(155.67\)
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:  $3^{1/2}103^{33/34}\approx 155.6670721921805$
Ramified primes:   \(3\), \(103\) 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:  \(\Q(\sqrt{309}) \)
$\card{ \Gal(K/\Q) }$:  $34$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(309=3\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{309}(1,·)$, $\chi_{309}(133,·)$, $\chi_{309}(134,·)$, $\chi_{309}(140,·)$, $\chi_{309}(13,·)$, $\chi_{309}(275,·)$, $\chi_{309}(34,·)$, $\chi_{309}(296,·)$, $\chi_{309}(169,·)$, $\chi_{309}(175,·)$, $\chi_{309}(176,·)$, $\chi_{309}(308,·)$, $\chi_{309}(184,·)$, $\chi_{309}(61,·)$, $\chi_{309}(64,·)$, $\chi_{309}(196,·)$, $\chi_{309}(197,·)$, $\chi_{309}(76,·)$, $\chi_{309}(79,·)$, $\chi_{309}(80,·)$, $\chi_{309}(209,·)$, $\chi_{309}(214,·)$, $\chi_{309}(89,·)$, $\chi_{309}(220,·)$, $\chi_{309}(95,·)$, $\chi_{309}(100,·)$, $\chi_{309}(229,·)$, $\chi_{309}(230,·)$, $\chi_{309}(233,·)$, $\chi_{309}(112,·)$, $\chi_{309}(113,·)$, $\chi_{309}(245,·)$, $\chi_{309}(248,·)$, $\chi_{309}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{47}a^{24}+\frac{16}{47}a^{23}+\frac{7}{47}a^{22}+\frac{22}{47}a^{21}-\frac{11}{47}a^{20}-\frac{13}{47}a^{17}-\frac{19}{47}a^{16}-\frac{17}{47}a^{15}+\frac{20}{47}a^{14}-\frac{9}{47}a^{13}+\frac{22}{47}a^{12}+\frac{5}{47}a^{11}+\frac{11}{47}a^{10}+\frac{6}{47}a^{9}-\frac{4}{47}a^{8}-\frac{17}{47}a^{7}-\frac{4}{47}a^{6}-\frac{22}{47}a^{5}+\frac{4}{47}a^{4}-\frac{19}{47}a^{3}+\frac{18}{47}a^{2}-\frac{1}{47}a$, $\frac{1}{47}a^{25}-\frac{14}{47}a^{23}+\frac{4}{47}a^{22}+\frac{13}{47}a^{21}-\frac{12}{47}a^{20}-\frac{13}{47}a^{18}+\frac{1}{47}a^{17}+\frac{5}{47}a^{16}+\frac{10}{47}a^{15}-\frac{22}{47}a^{13}-\frac{18}{47}a^{12}-\frac{22}{47}a^{11}+\frac{18}{47}a^{10}-\frac{6}{47}a^{9}-\frac{14}{47}a^{7}-\frac{5}{47}a^{6}-\frac{20}{47}a^{5}+\frac{11}{47}a^{4}-\frac{7}{47}a^{3}-\frac{7}{47}a^{2}+\frac{16}{47}a$, $\frac{1}{47}a^{26}-\frac{7}{47}a^{23}+\frac{17}{47}a^{22}+\frac{14}{47}a^{21}-\frac{13}{47}a^{20}-\frac{13}{47}a^{19}+\frac{1}{47}a^{18}+\frac{11}{47}a^{17}-\frac{21}{47}a^{16}-\frac{3}{47}a^{15}+\frac{23}{47}a^{14}-\frac{3}{47}a^{13}+\frac{4}{47}a^{12}-\frac{6}{47}a^{11}+\frac{7}{47}a^{10}-\frac{10}{47}a^{9}-\frac{23}{47}a^{8}-\frac{8}{47}a^{7}+\frac{18}{47}a^{6}-\frac{15}{47}a^{5}+\frac{2}{47}a^{4}+\frac{9}{47}a^{3}-\frac{14}{47}a^{2}-\frac{14}{47}a$, $\frac{1}{47}a^{27}-\frac{12}{47}a^{23}+\frac{16}{47}a^{22}+\frac{4}{47}a^{20}+\frac{1}{47}a^{19}+\frac{11}{47}a^{18}-\frac{18}{47}a^{17}+\frac{5}{47}a^{16}-\frac{2}{47}a^{15}-\frac{4}{47}a^{14}-\frac{12}{47}a^{13}+\frac{7}{47}a^{12}-\frac{5}{47}a^{11}+\frac{20}{47}a^{10}+\frac{19}{47}a^{9}+\frac{11}{47}a^{8}-\frac{7}{47}a^{7}+\frac{4}{47}a^{6}-\frac{11}{47}a^{5}-\frac{10}{47}a^{4}-\frac{6}{47}a^{3}+\frac{18}{47}a^{2}-\frac{7}{47}a$, $\frac{1}{47}a^{28}+\frac{20}{47}a^{23}-\frac{10}{47}a^{22}-\frac{14}{47}a^{21}+\frac{10}{47}a^{20}+\frac{11}{47}a^{19}-\frac{18}{47}a^{18}-\frac{10}{47}a^{17}+\frac{5}{47}a^{16}-\frac{20}{47}a^{15}-\frac{7}{47}a^{14}-\frac{7}{47}a^{13}-\frac{23}{47}a^{12}-\frac{14}{47}a^{11}+\frac{10}{47}a^{10}-\frac{11}{47}a^{9}-\frac{8}{47}a^{8}-\frac{12}{47}a^{7}-\frac{12}{47}a^{6}+\frac{8}{47}a^{5}-\frac{5}{47}a^{4}-\frac{22}{47}a^{3}+\frac{21}{47}a^{2}-\frac{12}{47}a$, $\frac{1}{12361}a^{29}-\frac{93}{12361}a^{28}-\frac{59}{12361}a^{27}+\frac{19}{12361}a^{26}-\frac{120}{12361}a^{25}+\frac{124}{12361}a^{24}+\frac{122}{12361}a^{23}+\frac{308}{12361}a^{22}+\frac{2494}{12361}a^{21}+\frac{1338}{12361}a^{20}-\frac{3321}{12361}a^{19}+\frac{1654}{12361}a^{18}-\frac{65}{12361}a^{17}+\frac{240}{12361}a^{16}+\frac{1061}{12361}a^{15}-\frac{1726}{12361}a^{14}-\frac{6135}{12361}a^{13}+\frac{3933}{12361}a^{12}+\frac{43}{263}a^{11}+\frac{5597}{12361}a^{10}-\frac{5485}{12361}a^{9}-\frac{4436}{12361}a^{8}-\frac{3564}{12361}a^{7}-\frac{5307}{12361}a^{6}+\frac{1090}{12361}a^{5}+\frac{3316}{12361}a^{4}-\frac{4748}{12361}a^{3}-\frac{2837}{12361}a^{2}-\frac{2359}{12361}a+\frac{76}{263}$, $\frac{1}{12361}a^{30}-\frac{29}{12361}a^{28}+\frac{55}{12361}a^{27}+\frac{69}{12361}a^{26}+\frac{10}{12361}a^{25}+\frac{82}{12361}a^{24}+\frac{345}{12361}a^{23}+\frac{6153}{12361}a^{22}+\frac{3418}{12361}a^{21}-\frac{1971}{12361}a^{20}-\frac{278}{12361}a^{19}-\frac{361}{12361}a^{18}+\frac{1559}{12361}a^{17}-\frac{5286}{12361}a^{16}+\frac{1741}{12361}a^{15}+\frac{1930}{12361}a^{14}+\frac{406}{12361}a^{13}-\frac{5670}{12361}a^{12}+\frac{245}{12361}a^{11}-\frac{5967}{12361}a^{10}+\frac{5410}{12361}a^{9}+\frac{3899}{12361}a^{8}-\frac{4590}{12361}a^{7}-\frac{6174}{12361}a^{6}-\frac{2355}{12361}a^{5}+\frac{5135}{12361}a^{4}+\frac{3751}{12361}a^{3}-\frac{833}{12361}a^{2}+\frac{108}{12361}a-\frac{33}{263}$, $\frac{1}{12361}a^{31}-\frac{12}{12361}a^{28}-\frac{64}{12361}a^{27}+\frac{35}{12361}a^{26}+\frac{21}{12361}a^{25}-\frac{4}{12361}a^{24}-\frac{2144}{12361}a^{23}+\frac{778}{12361}a^{22}-\frac{3811}{12361}a^{21}+\frac{6175}{12361}a^{20}+\frac{2481}{12361}a^{19}-\frac{708}{12361}a^{18}-\frac{596}{12361}a^{17}-\frac{3134}{12361}a^{16}+\frac{5610}{12361}a^{15}-\frac{4675}{12361}a^{14}+\frac{515}{12361}a^{13}+\frac{949}{12361}a^{12}+\frac{2672}{12361}a^{11}+\frac{5189}{12361}a^{10}+\frac{4738}{12361}a^{9}+\frac{2211}{12361}a^{8}-\frac{5119}{12361}a^{7}+\frac{5487}{12361}a^{6}+\frac{4922}{12361}a^{5}+\frac{5235}{12361}a^{4}+\frac{24}{263}a^{3}+\frac{417}{12361}a^{2}+\frac{2889}{12361}a+\frac{100}{263}$, $\frac{1}{580967}a^{32}-\frac{8}{580967}a^{31}-\frac{18}{580967}a^{30}+\frac{12}{580967}a^{29}-\frac{889}{580967}a^{28}+\frac{2349}{580967}a^{27}-\frac{67}{12361}a^{26}-\frac{2180}{580967}a^{25}-\frac{2190}{580967}a^{24}-\frac{70038}{580967}a^{23}-\frac{121287}{580967}a^{22}-\frac{150157}{580967}a^{21}-\frac{145019}{580967}a^{20}-\frac{289876}{580967}a^{19}+\frac{252983}{580967}a^{18}-\frac{162644}{580967}a^{17}-\frac{208995}{580967}a^{16}-\frac{221382}{580967}a^{15}-\frac{78751}{580967}a^{14}-\frac{30953}{580967}a^{13}+\frac{227300}{580967}a^{12}+\frac{2133}{580967}a^{11}+\frac{61362}{580967}a^{10}+\frac{219733}{580967}a^{9}+\frac{253696}{580967}a^{8}+\frac{21168}{580967}a^{7}+\frac{211998}{580967}a^{6}+\frac{126196}{580967}a^{5}-\frac{36766}{580967}a^{4}-\frac{253723}{580967}a^{3}-\frac{91150}{580967}a^{2}+\frac{142107}{580967}a-\frac{1012}{12361}$, $\frac{1}{34\!\cdots\!47}a^{33}+\frac{28\!\cdots\!14}{34\!\cdots\!47}a^{32}-\frac{72\!\cdots\!35}{34\!\cdots\!47}a^{31}+\frac{12\!\cdots\!79}{34\!\cdots\!47}a^{30}-\frac{73\!\cdots\!06}{34\!\cdots\!47}a^{29}-\frac{20\!\cdots\!29}{34\!\cdots\!47}a^{28}-\frac{77\!\cdots\!15}{34\!\cdots\!47}a^{27}+\frac{70\!\cdots\!71}{34\!\cdots\!47}a^{26}-\frac{99\!\cdots\!79}{34\!\cdots\!47}a^{25}-\frac{27\!\cdots\!59}{34\!\cdots\!47}a^{24}+\frac{79\!\cdots\!31}{34\!\cdots\!47}a^{23}+\frac{15\!\cdots\!99}{34\!\cdots\!47}a^{22}-\frac{10\!\cdots\!56}{34\!\cdots\!47}a^{21}-\frac{19\!\cdots\!20}{34\!\cdots\!47}a^{20}+\frac{79\!\cdots\!74}{34\!\cdots\!47}a^{19}-\frac{10\!\cdots\!12}{34\!\cdots\!47}a^{18}+\frac{25\!\cdots\!15}{34\!\cdots\!47}a^{17}-\frac{12\!\cdots\!90}{34\!\cdots\!47}a^{16}-\frac{91\!\cdots\!53}{34\!\cdots\!47}a^{15}-\frac{37\!\cdots\!39}{34\!\cdots\!47}a^{14}-\frac{67\!\cdots\!15}{34\!\cdots\!47}a^{13}-\frac{36\!\cdots\!40}{34\!\cdots\!47}a^{12}+\frac{23\!\cdots\!04}{34\!\cdots\!47}a^{11}-\frac{17\!\cdots\!49}{34\!\cdots\!47}a^{10}-\frac{13\!\cdots\!54}{34\!\cdots\!47}a^{9}+\frac{11\!\cdots\!58}{34\!\cdots\!47}a^{8}+\frac{29\!\cdots\!52}{34\!\cdots\!47}a^{7}-\frac{14\!\cdots\!06}{34\!\cdots\!47}a^{6}+\frac{13\!\cdots\!73}{34\!\cdots\!47}a^{5}+\frac{80\!\cdots\!13}{34\!\cdots\!47}a^{4}+\frac{15\!\cdots\!60}{34\!\cdots\!47}a^{3}-\frac{14\!\cdots\!99}{34\!\cdots\!47}a^{2}-\frac{48\!\cdots\!85}{34\!\cdots\!47}a+\frac{11\!\cdots\!90}{73\!\cdots\!01}$ 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

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:  $33$
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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967)
 
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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967, 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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967);
 
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^34 - x^33 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967);
 
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

$C_{34}$ (as 34T1):

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 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$

Intermediate fields

\(\Q(\sqrt{309}) \), 17.17.160470643909878751793805444097921.1

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 $34$ R $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $34$ $34$ $34$ $34$ $34$ $34$ ${\href{/padicField/47.1.0.1}{1} }^{34}$ $17^{2}$ $34$

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
\(3\) Copy content Toggle raw display Deg $34$$2$$17$$17$
\(103\) Copy content Toggle raw display Deg $34$$34$$1$$33$