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Properties

Label 45.45.112...281.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.124\times 10^{107}$
Root discriminant \(239.28\)
Ramified prime $271$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997)
 
gp: K = bnfinit(y^45 - y^44 - 132*y^43 + 301*y^42 + 7581*y^41 - 27065*y^40 - 236809*y^39 + 1229173*y^38 + 3960874*y^37 - 33065607*y^36 - 18797995*y^35 + 555716542*y^34 - 619527603*y^33 - 5732299123*y^32 + 15235581842*y^31 + 30609318626*y^30 - 168897578817*y^29 + 17112929652*y^28 + 1054352970448*y^27 - 1501837759883*y^26 - 3320354588485*y^25 + 10782132568732*y^24 - 107051646226*y^23 - 36936809485370*y^22 + 41830258905956*y^21 + 53083505529011*y^20 - 152445251342987*y^19 + 42842896041923*y^18 + 229594603485553*y^17 - 272222797945867*y^16 - 70739760279828*y^15 + 364126674087072*y^14 - 209159107972130*y^13 - 142016333800362*y^12 + 231101895832035*y^11 - 63870399263881*y^10 - 62082302112786*y^9 + 49149720219336*y^8 - 3670701881262*y^7 - 8040338248733*y^6 + 2565873490427*y^5 + 306896523050*y^4 - 230447557112*y^3 + 11171763392*y^2 + 5932497375*y - 637239997, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997)
 

\( x^{45} - x^{44} - 132 x^{43} + 301 x^{42} + 7581 x^{41} - 27065 x^{40} - 236809 x^{39} + \cdots - 637239997 \) Copy content Toggle raw display

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(112\!\cdots\!281\) \(\medspace = 271^{44}\) 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:  \(239.28\)
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:  $271^{44/45}\approx 239.27829566371238$
Ramified primes:   \(271\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $45$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(271\)
Dirichlet character group:    $\lbrace$$\chi_{271}(1,·)$, $\chi_{271}(258,·)$, $\chi_{271}(8,·)$, $\chi_{271}(9,·)$, $\chi_{271}(10,·)$, $\chi_{271}(139,·)$, $\chi_{271}(268,·)$, $\chi_{271}(141,·)$, $\chi_{271}(148,·)$, $\chi_{271}(154,·)$, $\chi_{271}(28,·)$, $\chi_{271}(31,·)$, $\chi_{271}(34,·)$, $\chi_{271}(35,·)$, $\chi_{271}(166,·)$, $\chi_{271}(39,·)$, $\chi_{271}(41,·)$, $\chi_{271}(44,·)$, $\chi_{271}(178,·)$, $\chi_{271}(55,·)$, $\chi_{271}(57,·)$, $\chi_{271}(87,·)$, $\chi_{271}(187,·)$, $\chi_{271}(64,·)$, $\chi_{271}(69,·)$, $\chi_{271}(72,·)$, $\chi_{271}(119,·)$, $\chi_{271}(79,·)$, $\chi_{271}(80,·)$, $\chi_{271}(81,·)$, $\chi_{271}(185,·)$, $\chi_{271}(169,·)$, $\chi_{271}(90,·)$, $\chi_{271}(224,·)$, $\chi_{271}(98,·)$, $\chi_{271}(100,·)$, $\chi_{271}(106,·)$, $\chi_{271}(167,·)$, $\chi_{271}(241,·)$, $\chi_{271}(242,·)$, $\chi_{271}(244,·)$, $\chi_{271}(247,·)$, $\chi_{271}(248,·)$, $\chi_{271}(252,·)$, $\chi_{271}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{29}a^{24}+\frac{1}{29}a^{23}-\frac{4}{29}a^{22}+\frac{5}{29}a^{21}+\frac{7}{29}a^{20}-\frac{13}{29}a^{19}+\frac{11}{29}a^{18}-\frac{10}{29}a^{17}-\frac{6}{29}a^{16}-\frac{11}{29}a^{15}+\frac{9}{29}a^{13}-\frac{13}{29}a^{12}-\frac{3}{29}a^{11}+\frac{6}{29}a^{10}-\frac{14}{29}a^{9}+\frac{7}{29}a^{8}+\frac{5}{29}a^{7}-\frac{1}{29}a^{6}+\frac{5}{29}a^{5}+\frac{4}{29}a^{4}-\frac{8}{29}a^{3}-\frac{12}{29}a^{2}+\frac{5}{29}a$, $\frac{1}{29}a^{25}-\frac{5}{29}a^{23}+\frac{9}{29}a^{22}+\frac{2}{29}a^{21}+\frac{9}{29}a^{20}-\frac{5}{29}a^{19}+\frac{8}{29}a^{18}+\frac{4}{29}a^{17}-\frac{5}{29}a^{16}+\frac{11}{29}a^{15}+\frac{9}{29}a^{14}+\frac{7}{29}a^{13}+\frac{10}{29}a^{12}+\frac{9}{29}a^{11}+\frac{9}{29}a^{10}-\frac{8}{29}a^{9}-\frac{2}{29}a^{8}-\frac{6}{29}a^{7}+\frac{6}{29}a^{6}-\frac{1}{29}a^{5}-\frac{12}{29}a^{4}-\frac{4}{29}a^{3}-\frac{12}{29}a^{2}-\frac{5}{29}a$, $\frac{1}{29}a^{26}+\frac{14}{29}a^{23}+\frac{11}{29}a^{22}+\frac{5}{29}a^{21}+\frac{1}{29}a^{20}+\frac{1}{29}a^{19}+\frac{1}{29}a^{18}+\frac{3}{29}a^{17}+\frac{10}{29}a^{16}+\frac{12}{29}a^{15}+\frac{7}{29}a^{14}-\frac{3}{29}a^{13}+\frac{2}{29}a^{12}-\frac{6}{29}a^{11}-\frac{7}{29}a^{10}-\frac{14}{29}a^{9}+\frac{2}{29}a^{7}-\frac{6}{29}a^{6}+\frac{13}{29}a^{5}-\frac{13}{29}a^{4}+\frac{6}{29}a^{3}-\frac{7}{29}a^{2}-\frac{4}{29}a$, $\frac{1}{29}a^{27}-\frac{3}{29}a^{23}+\frac{3}{29}a^{22}-\frac{11}{29}a^{21}-\frac{10}{29}a^{20}+\frac{9}{29}a^{19}-\frac{6}{29}a^{18}+\frac{5}{29}a^{17}+\frac{9}{29}a^{16}-\frac{13}{29}a^{15}-\frac{3}{29}a^{14}-\frac{8}{29}a^{13}+\frac{2}{29}a^{12}+\frac{6}{29}a^{11}-\frac{11}{29}a^{10}-\frac{7}{29}a^{9}-\frac{9}{29}a^{8}+\frac{11}{29}a^{7}-\frac{2}{29}a^{6}+\frac{4}{29}a^{5}+\frac{8}{29}a^{4}-\frac{11}{29}a^{3}-\frac{10}{29}a^{2}-\frac{12}{29}a$, $\frac{1}{29}a^{28}+\frac{6}{29}a^{23}+\frac{6}{29}a^{22}+\frac{5}{29}a^{21}+\frac{1}{29}a^{20}+\frac{13}{29}a^{19}+\frac{9}{29}a^{18}+\frac{8}{29}a^{17}-\frac{2}{29}a^{16}-\frac{7}{29}a^{15}-\frac{8}{29}a^{14}-\frac{4}{29}a^{12}+\frac{9}{29}a^{11}+\frac{11}{29}a^{10}+\frac{7}{29}a^{9}+\frac{3}{29}a^{8}+\frac{13}{29}a^{7}+\frac{1}{29}a^{6}-\frac{6}{29}a^{5}+\frac{1}{29}a^{4}-\frac{5}{29}a^{3}+\frac{10}{29}a^{2}-\frac{14}{29}a$, $\frac{1}{29}a^{29}-\frac{1}{29}a$, $\frac{1}{29}a^{30}-\frac{1}{29}a^{2}$, $\frac{1}{29}a^{31}-\frac{1}{29}a^{3}$, $\frac{1}{29}a^{32}-\frac{1}{29}a^{4}$, $\frac{1}{29}a^{33}-\frac{1}{29}a^{5}$, $\frac{1}{29}a^{34}-\frac{1}{29}a^{6}$, $\frac{1}{29}a^{35}-\frac{1}{29}a^{7}$, $\frac{1}{29}a^{36}-\frac{1}{29}a^{8}$, $\frac{1}{29}a^{37}-\frac{1}{29}a^{9}$, $\frac{1}{29}a^{38}-\frac{1}{29}a^{10}$, $\frac{1}{841}a^{39}+\frac{1}{841}a^{38}-\frac{8}{841}a^{36}+\frac{12}{841}a^{35}-\frac{13}{841}a^{34}+\frac{9}{841}a^{33}+\frac{12}{841}a^{32}-\frac{8}{841}a^{31}+\frac{1}{841}a^{30}+\frac{11}{841}a^{29}-\frac{11}{841}a^{27}-\frac{11}{841}a^{26}+\frac{2}{841}a^{25}-\frac{2}{841}a^{24}-\frac{394}{841}a^{23}+\frac{365}{841}a^{22}+\frac{205}{841}a^{21}+\frac{277}{841}a^{20}-\frac{297}{841}a^{19}+\frac{252}{841}a^{18}+\frac{114}{841}a^{17}+\frac{112}{841}a^{16}+\frac{84}{841}a^{15}-\frac{84}{841}a^{14}-\frac{173}{841}a^{13}-\frac{27}{841}a^{12}+\frac{313}{841}a^{11}-\frac{3}{29}a^{10}-\frac{250}{841}a^{9}-\frac{346}{841}a^{8}-\frac{32}{841}a^{7}+\frac{57}{841}a^{6}-\frac{324}{841}a^{5}-\frac{337}{841}a^{4}+\frac{71}{841}a^{3}-\frac{162}{841}a^{2}-\frac{6}{29}a$, $\frac{1}{841}a^{40}-\frac{1}{841}a^{38}-\frac{8}{841}a^{37}-\frac{9}{841}a^{36}+\frac{4}{841}a^{35}-\frac{7}{841}a^{34}+\frac{3}{841}a^{33}+\frac{9}{841}a^{32}+\frac{9}{841}a^{31}+\frac{10}{841}a^{30}-\frac{11}{841}a^{29}-\frac{11}{841}a^{28}+\frac{13}{841}a^{26}-\frac{4}{841}a^{25}+\frac{14}{841}a^{24}+\frac{324}{841}a^{23}-\frac{102}{841}a^{22}+\frac{420}{841}a^{21}-\frac{255}{841}a^{20}+\frac{317}{841}a^{19}+\frac{123}{841}a^{18}+\frac{143}{841}a^{17}+\frac{59}{841}a^{16}+\frac{412}{841}a^{15}-\frac{89}{841}a^{14}-\frac{405}{841}a^{13}+\frac{108}{841}a^{12}+\frac{64}{841}a^{11}-\frac{250}{841}a^{10}+\frac{107}{841}a^{9}-\frac{179}{841}a^{8}+\frac{408}{841}a^{7}+\frac{83}{841}a^{6}+\frac{335}{841}a^{5}+\frac{321}{841}a^{4}-\frac{117}{841}a^{3}+\frac{162}{841}a^{2}-\frac{11}{29}a$, $\frac{1}{841}a^{41}-\frac{7}{841}a^{38}-\frac{9}{841}a^{37}-\frac{4}{841}a^{36}+\frac{5}{841}a^{35}-\frac{10}{841}a^{34}-\frac{11}{841}a^{33}-\frac{8}{841}a^{32}+\frac{2}{841}a^{31}-\frac{10}{841}a^{30}+\frac{2}{841}a^{27}+\frac{14}{841}a^{26}-\frac{13}{841}a^{25}+\frac{3}{841}a^{24}-\frac{264}{841}a^{23}-\frac{404}{841}a^{22}+\frac{124}{841}a^{21}-\frac{189}{841}a^{20}-\frac{2}{29}a^{19}+\frac{47}{841}a^{18}-\frac{30}{841}a^{17}+\frac{350}{841}a^{16}+\frac{169}{841}a^{15}+\frac{294}{841}a^{14}+\frac{138}{841}a^{13}-\frac{253}{841}a^{12}-\frac{256}{841}a^{11}+\frac{165}{841}a^{10}-\frac{342}{841}a^{9}+\frac{410}{841}a^{8}+\frac{370}{841}a^{7}+\frac{363}{841}a^{6}-\frac{322}{841}a^{5}-\frac{48}{841}a^{4}-\frac{289}{841}a^{3}+\frac{128}{841}a^{2}-\frac{2}{29}a$, $\frac{1}{841}a^{42}-\frac{2}{841}a^{38}-\frac{4}{841}a^{37}+\frac{7}{841}a^{36}-\frac{13}{841}a^{35}+\frac{14}{841}a^{34}-\frac{3}{841}a^{33}-\frac{1}{841}a^{32}-\frac{8}{841}a^{31}+\frac{7}{841}a^{30}-\frac{10}{841}a^{29}+\frac{2}{841}a^{28}-\frac{5}{841}a^{27}-\frac{3}{841}a^{26}-\frac{12}{841}a^{25}+\frac{12}{841}a^{24}-\frac{1}{841}a^{23}-\frac{134}{841}a^{22}-\frac{88}{841}a^{21}-\frac{207}{841}a^{20}-\frac{2}{841}a^{19}+\frac{226}{841}a^{18}+\frac{365}{841}a^{17}-\frac{91}{841}a^{16}+\frac{186}{841}a^{15}-\frac{276}{841}a^{14}+\frac{218}{841}a^{13}-\frac{10}{841}a^{12}+\frac{210}{841}a^{11}+\frac{122}{841}a^{10}-\frac{64}{841}a^{9}+\frac{297}{841}a^{8}+\frac{139}{841}a^{7}-\frac{300}{841}a^{6}-\frac{257}{841}a^{5}-\frac{38}{841}a^{4}-\frac{71}{841}a^{3}+\frac{374}{841}a^{2}+\frac{9}{29}a$, $\frac{1}{16\!\cdots\!91}a^{43}+\frac{6626171496925}{16\!\cdots\!91}a^{42}+\frac{4884261094287}{16\!\cdots\!91}a^{41}+\frac{1625691147042}{16\!\cdots\!91}a^{40}+\frac{6218276866477}{16\!\cdots\!91}a^{39}+\frac{29337529532410}{16\!\cdots\!91}a^{38}-\frac{226142381910548}{16\!\cdots\!91}a^{37}-\frac{81549296544456}{16\!\cdots\!91}a^{36}+\frac{231208942539810}{16\!\cdots\!91}a^{35}+\frac{229416032903837}{16\!\cdots\!91}a^{34}-\frac{44967368787353}{16\!\cdots\!91}a^{33}+\frac{87382394642794}{16\!\cdots\!91}a^{32}+\frac{213837181029428}{16\!\cdots\!91}a^{31}-\frac{165541714854839}{16\!\cdots\!91}a^{30}+\frac{221477940113616}{16\!\cdots\!91}a^{29}-\frac{272381537118142}{16\!\cdots\!91}a^{28}+\frac{8392212270605}{560903106039379}a^{27}-\frac{275532105396530}{16\!\cdots\!91}a^{26}-\frac{141438453858530}{16\!\cdots\!91}a^{25}-\frac{137404490719787}{16\!\cdots\!91}a^{24}-\frac{10\!\cdots\!77}{16\!\cdots\!91}a^{23}+\frac{55\!\cdots\!79}{16\!\cdots\!91}a^{22}-\frac{16\!\cdots\!40}{16\!\cdots\!91}a^{21}-\frac{184986981206822}{16\!\cdots\!91}a^{20}-\frac{73\!\cdots\!28}{16\!\cdots\!91}a^{19}-\frac{55\!\cdots\!12}{16\!\cdots\!91}a^{18}-\frac{11\!\cdots\!11}{16\!\cdots\!91}a^{17}+\frac{62\!\cdots\!46}{16\!\cdots\!91}a^{16}-\frac{71\!\cdots\!04}{16\!\cdots\!91}a^{15}+\frac{43\!\cdots\!43}{16\!\cdots\!91}a^{14}+\frac{44\!\cdots\!43}{16\!\cdots\!91}a^{13}+\frac{71\!\cdots\!76}{16\!\cdots\!91}a^{12}+\frac{59\!\cdots\!92}{16\!\cdots\!91}a^{11}-\frac{403970397352969}{16\!\cdots\!91}a^{10}-\frac{18\!\cdots\!91}{16\!\cdots\!91}a^{9}+\frac{59\!\cdots\!22}{16\!\cdots\!91}a^{8}+\frac{892937494708028}{16\!\cdots\!91}a^{7}+\frac{74\!\cdots\!70}{16\!\cdots\!91}a^{6}-\frac{26\!\cdots\!38}{16\!\cdots\!91}a^{5}-\frac{74220598602075}{560903106039379}a^{4}+\frac{57\!\cdots\!45}{16\!\cdots\!91}a^{3}+\frac{41\!\cdots\!83}{16\!\cdots\!91}a^{2}-\frac{236900816131711}{560903106039379}a+\frac{4512539}{25526003}$, $\frac{1}{78\!\cdots\!63}a^{44}-\frac{23\!\cdots\!61}{78\!\cdots\!63}a^{43}-\frac{35\!\cdots\!93}{78\!\cdots\!63}a^{42}-\frac{29\!\cdots\!64}{78\!\cdots\!63}a^{41}+\frac{44\!\cdots\!96}{78\!\cdots\!63}a^{40}+\frac{56\!\cdots\!56}{78\!\cdots\!63}a^{39}-\frac{51\!\cdots\!54}{78\!\cdots\!63}a^{38}+\frac{42\!\cdots\!63}{78\!\cdots\!63}a^{37}+\frac{56\!\cdots\!79}{78\!\cdots\!63}a^{36}+\frac{43\!\cdots\!42}{78\!\cdots\!63}a^{35}-\frac{77\!\cdots\!48}{78\!\cdots\!63}a^{34}-\frac{11\!\cdots\!99}{78\!\cdots\!63}a^{33}+\frac{28\!\cdots\!47}{78\!\cdots\!63}a^{32}+\frac{88\!\cdots\!91}{78\!\cdots\!63}a^{31}-\frac{14\!\cdots\!04}{78\!\cdots\!63}a^{30}+\frac{12\!\cdots\!66}{78\!\cdots\!63}a^{29}+\frac{10\!\cdots\!51}{78\!\cdots\!63}a^{28}+\frac{12\!\cdots\!93}{78\!\cdots\!63}a^{27}-\frac{92\!\cdots\!16}{78\!\cdots\!63}a^{26}-\frac{13\!\cdots\!71}{78\!\cdots\!63}a^{25}+\frac{29\!\cdots\!82}{78\!\cdots\!63}a^{24}-\frac{17\!\cdots\!77}{78\!\cdots\!63}a^{23}-\frac{10\!\cdots\!91}{78\!\cdots\!63}a^{22}+\frac{21\!\cdots\!12}{78\!\cdots\!63}a^{21}+\frac{10\!\cdots\!81}{27\!\cdots\!47}a^{20}-\frac{31\!\cdots\!91}{78\!\cdots\!63}a^{19}-\frac{36\!\cdots\!67}{78\!\cdots\!63}a^{18}-\frac{24\!\cdots\!84}{78\!\cdots\!63}a^{17}-\frac{22\!\cdots\!65}{78\!\cdots\!63}a^{16}-\frac{15\!\cdots\!28}{78\!\cdots\!63}a^{15}-\frac{25\!\cdots\!15}{78\!\cdots\!63}a^{14}+\frac{31\!\cdots\!92}{78\!\cdots\!63}a^{13}+\frac{95\!\cdots\!54}{78\!\cdots\!63}a^{12}-\frac{19\!\cdots\!06}{78\!\cdots\!63}a^{11}+\frac{21\!\cdots\!70}{78\!\cdots\!63}a^{10}+\frac{27\!\cdots\!92}{78\!\cdots\!63}a^{9}-\frac{39\!\cdots\!78}{78\!\cdots\!63}a^{8}-\frac{27\!\cdots\!96}{78\!\cdots\!63}a^{7}-\frac{56\!\cdots\!15}{78\!\cdots\!63}a^{6}+\frac{31\!\cdots\!85}{78\!\cdots\!63}a^{5}-\frac{36\!\cdots\!33}{78\!\cdots\!63}a^{4}-\frac{21\!\cdots\!46}{78\!\cdots\!63}a^{3}-\frac{28\!\cdots\!43}{78\!\cdots\!63}a^{2}-\frac{19\!\cdots\!79}{27\!\cdots\!47}a+\frac{57\!\cdots\!18}{12\!\cdots\!79}$ 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:  $44$
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^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997)
 
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^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997, 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^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997);
 
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^45 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997);
 
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

$C_{45}$ (as 45T1):

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 45
The 45 conjugacy class representatives for $C_{45}$
Character table for $C_{45}$

Intermediate fields

3.3.73441.1, 5.5.5393580481.1, 9.9.29090710405024191361.1, 15.15.11523119672512394327137541804059681.1

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 $45$ ${\href{/padicField/3.5.0.1}{5} }^{9}$ ${\href{/padicField/5.9.0.1}{9} }^{5}$ $45$ $45$ ${\href{/padicField/13.3.0.1}{3} }^{15}$ $45$ ${\href{/padicField/19.5.0.1}{5} }^{9}$ ${\href{/padicField/23.3.0.1}{3} }^{15}$ ${\href{/padicField/29.1.0.1}{1} }^{45}$ $15^{3}$ $45$ $15^{3}$ $45$ $15^{3}$ $45$ $45$

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
\(271\) Copy content Toggle raw display Deg $45$$45$$1$$44$