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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(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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-637239997, 5932497375, 11171763392, -230447557112, 306896523050, 2565873490427, -8040338248733, -3670701881262, 49149720219336, -62082302112786, -63870399263881, 231101895832035, -142016333800362, -209159107972130, 364126674087072, -70739760279828, -272222797945867, 229594603485553, 42842896041923, -152445251342987, 53083505529011, 41830258905956, -36936809485370, -107051646226, 10782132568732, -3320354588485, -1501837759883, 1054352970448, 17112929652, -168897578817, 30609318626, 15235581842, -5732299123, -619527603, 555716542, -18797995, -33065607, 3960874, 1229173, -236809, -27065, 7581, 301, -132, -1, 1]);
 

\( x^{45} - x^{44} - 132 x^{43} + 301 x^{42} + 7581 x^{41} - 27065 x^{40} - 236809 x^{39} + 1229173 x^{38} + 3960874 x^{37} - 33065607 x^{36} - 18797995 x^{35} + 555716542 x^{34} - 619527603 x^{33} - 5732299123 x^{32} + 15235581842 x^{31} + 30609318626 x^{30} - 168897578817 x^{29} + 17112929652 x^{28} + 1054352970448 x^{27} - 1501837759883 x^{26} - 3320354588485 x^{25} + 10782132568732 x^{24} - 107051646226 x^{23} - 36936809485370 x^{22} + 41830258905956 x^{21} + 53083505529011 x^{20} - 152445251342987 x^{19} + 42842896041923 x^{18} + 229594603485553 x^{17} - 272222797945867 x^{16} - 70739760279828 x^{15} + 364126674087072 x^{14} - 209159107972130 x^{13} - 142016333800362 x^{12} + 231101895832035 x^{11} - 63870399263881 x^{10} - 62082302112786 x^{9} + 49149720219336 x^{8} - 3670701881262 x^{7} - 8040338248733 x^{6} + 2565873490427 x^{5} + 306896523050 x^{4} - 230447557112 x^{3} + 11171763392 x^{2} + 5932497375 x - 637239997 \)

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(112\!\cdots\!281\)\(\medspace = 271^{44}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $239.28$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $271$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $45$
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}{16266190075141991} a^{43} + \frac{6626171496925}{16266190075141991} a^{42} + \frac{4884261094287}{16266190075141991} a^{41} + \frac{1625691147042}{16266190075141991} a^{40} + \frac{6218276866477}{16266190075141991} a^{39} + \frac{29337529532410}{16266190075141991} a^{38} - \frac{226142381910548}{16266190075141991} a^{37} - \frac{81549296544456}{16266190075141991} a^{36} + \frac{231208942539810}{16266190075141991} a^{35} + \frac{229416032903837}{16266190075141991} a^{34} - \frac{44967368787353}{16266190075141991} a^{33} + \frac{87382394642794}{16266190075141991} a^{32} + \frac{213837181029428}{16266190075141991} a^{31} - \frac{165541714854839}{16266190075141991} a^{30} + \frac{221477940113616}{16266190075141991} a^{29} - \frac{272381537118142}{16266190075141991} a^{28} + \frac{8392212270605}{560903106039379} a^{27} - \frac{275532105396530}{16266190075141991} a^{26} - \frac{141438453858530}{16266190075141991} a^{25} - \frac{137404490719787}{16266190075141991} a^{24} - \frac{1077421407501977}{16266190075141991} a^{23} + \frac{5575997017336679}{16266190075141991} a^{22} - \frac{1624506534594840}{16266190075141991} a^{21} - \frac{184986981206822}{16266190075141991} a^{20} - \frac{7375434432613228}{16266190075141991} a^{19} - \frac{5556192505072412}{16266190075141991} a^{18} - \frac{1131250336404411}{16266190075141991} a^{17} + \frac{6208692957021446}{16266190075141991} a^{16} - \frac{7166136545857904}{16266190075141991} a^{15} + \frac{4349910761640243}{16266190075141991} a^{14} + \frac{4436119341993243}{16266190075141991} a^{13} + \frac{7120146691637976}{16266190075141991} a^{12} + \frac{5953104919832992}{16266190075141991} a^{11} - \frac{403970397352969}{16266190075141991} a^{10} - \frac{1810022455821091}{16266190075141991} a^{9} + \frac{5963007601683122}{16266190075141991} a^{8} + \frac{892937494708028}{16266190075141991} a^{7} + \frac{7425122301788170}{16266190075141991} a^{6} - \frac{2610830563813838}{16266190075141991} a^{5} - \frac{74220598602075}{560903106039379} a^{4} + \frac{5771780051248445}{16266190075141991} a^{3} + \frac{4161379273677483}{16266190075141991} a^{2} - \frac{236900816131711}{560903106039379} a + \frac{4512539}{25526003}$, $\frac{1}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{44} - \frac{232244162362480851751953354117362196407043322870231280829609559959528154104730507606857772765199215290104661}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{43} - \frac{3586489006460970767881538567415238377086439892442687627645708073365581418628090914369993227348063287655384865795974297093}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{42} - \frac{2976721675995188796461489580351411024990714444954444669647764819172414759089372730550105352142008651864112943559856387064}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{41} + \frac{4424558815497617564233119324551829020947393237928733262596770725119017352182920055962224453528156583542213365068637068796}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{40} + \frac{568775530256452677696186396601491057531086450583296620389096757788756494386850667540218673841883939870876596847303829356}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{39} - \frac{51529520587493072203819047482144159674369459167001799673154061189125630445999717544716341441998448303648319032090480511954}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{38} + \frac{42877858832614411695978478005742276001442865051850318713565028603774860698202129568616893268909014864447757701901635669363}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{37} + \frac{56097626312122156992705788901664533571086426124350640785442800136284397607602228966371155115409253315899376486785664772379}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{36} + \frac{43014303921575879672466554678237345032033494896697914056440071411220350700224570336806667224944901695123811670297321075442}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{35} - \frac{77863780777714465813022156464731483972281137158220232418809306165807455960159395970973230207450944830756493718357867906048}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{34} - \frac{110162399645257464575197243401247075595348156918974309971285320543952101321535811586911240283402010491280395630334055168199}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{33} + \frac{28246402762921746282925119590066826109117785824835399814437426082147219305678192575133285651851805531816197125742713858447}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{32} + \frac{88414701450789595341739755806020273502163477237045457939489344005127843191320836293404127718165903095971006736149804765591}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{31} - \frac{14596263083721139644321842296273653347266484827151506336151328566619271672530488933933996789303420442304075125213861890004}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{30} + \frac{129798181695985099133488717738528252175552351927483630780955596397337807450135158058200433869446078524970297934015868914866}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{29} + \frac{10545934644703811248706526010126739749352859456215437597886561595532410060138499998615455076550867577946372002232378739951}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{28} + \frac{122084117250076080666251638008495656753864236449613064000113267752638847406308764011469202169578733106603967519409492474093}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{27} - \frac{92125210316341919927164271147499533174165861289761097221412185507002690303188660127739110942772103665222821822831266395216}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{26} - \frac{130722310575774535714627984464194825559477287020244309893808556906228631234605786548404331525854251280844107201938318810271}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{25} + \frac{2971072716535659127233104914494578527380895835034448376447972800629029168154622818988812614321228172305929146130995621282}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{24} - \frac{1724604865383471290822031469162424971564915927741980332170543704297017317814512510529818949396265257349963929737662461090477}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{23} - \frac{1099168224529901896362378265452712260158152549355351882607783747171252004341806714829795075751499918062002984773769401588391}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{22} + \frac{2139203148264569845870195553358093598558660826423240923634753263804822003617266123259829556489325740996916514793921745479312}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{21} + \frac{102634878279608680946874798826110227672831363589976329774854805615985575089349754097207602929044032047703897204342542379681}{270500291907461504376048507453389399601764977025879337909788828548428173061806935907551775925683229719607804031700185492147} a^{20} - \frac{313406576553491204242691594869517389707982472631398367787595418814559276063727938346225007476411363175308628637024389415891}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{19} - \frac{3619754198229869828855064727250877925631228489497969256683545218571623860667604792473079315647882905233338320115885836634267}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{18} - \frac{2461639113357254285687585320218182636179251880452153351399956174867219826617087054850441391888220581757778838521529053750484}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{17} - \frac{2237584785624898291479510046915549010747184685184514258503215898150006633677505639937717272042044907836528990447923585389465}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{16} - \frac{1536817537163341779352898244356608133548740557659349781745145668973269997653212405851326927493201083422575039826214575642328}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{15} - \frac{2572903942127577211211272769099019944113982251428462894074025179458868057191938938112896045086868272353982366527439210361115}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{14} + \frac{3188755583890164534041650466226137540654576803740189938065638746902457581785342932870969517034919347107924305798790871046992}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{13} + \frac{954717004614757053065203847651754716843293051416111650104629062830319261113419279357555990808119036564999744202713551212254}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{12} - \frac{1977370890945823096002498371608778858765791847302791946460247607141497758004828166925827794998043091811229482168902956986606}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{11} + \frac{2136596817109244294560522489290778703213001008748676962922316553760999512528376115592474540210129646405456528383916065520470}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{10} + \frac{2732857979977949028828541287171098670077709296945216028756994244206421347582426086051878051324783267918975668264950383999092}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{9} - \frac{3908364639636278123345863195339789639064164453903587712200423106510801482675728637548155295800945626124881450067479218849478}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{8} - \frac{2766295793008375893436421269926762547895166438705950440568396527897536175498859634958689329300841357841327132113611149093996}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{7} - \frac{563486891128741174534205946290122468149185951578165655074303837525146098784442354635176935619400312963760459149633432019315}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{6} + \frac{3162626989686410682548998403991199267460222316064857067217393447028210912226136118404515406691493434542737628103661979512085}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{5} - \frac{3699175913800473638385094096836532373691067150970734957973138053816359082756858804041248363593182706656352283163175073310533}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{4} - \frac{2183288063806407619041057157593886036584032582675799245058366827275914791332965593038070979842438293439206885867814548959946}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{3} - \frac{2890869311742537527747410778498474391651685275485541972429668106830058992154957394898309324926874285472467911763520623281443}{7844508465316383626905406716148292588451184333750500799383876027904417018792401141319001501844813661868626316919305379272263} a^{2} - \frac{19745412513705280572396752649891779641964750930270137928560679129800626952146393084042059608050326426029361921438490418279}{270500291907461504376048507453389399601764977025879337909788828548428173061806935907551775925683229719607804031700185492147} a + \frac{5710929583495456401017384206020911548427565702842979367412092955126965502406662405317459048313545220157901358011618}{12310131978919684206365669661737024627553603377709043582497970584706435209515577756992239615877114602818357487562579}$

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $44$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 45
The 45 conjugacy class representatives for $C_{45}$
Character table for $C_{45}$ is not computed

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.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $45$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{5}$ $45$ $45$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{15}$ $45$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{15}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{45}$ $15^{3}$ $45$ $15^{3}$ $45$ $15^{3}$ $45$ $45$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
271Data not computed