Properties

Label 45.45.217...129.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.172\times 10^{100}$
Root discriminant $169.71$
Ramified primes $7, 61$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)
 
gp: K = bnfinit(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6089, -9309, 2457345, -5291206, -288043322, 1537215457, 5258463166, -45168898819, 14178066293, 358975449036, -396963769511, -1316532612469, 2011525458240, 2589633668016, -5144196139934, -2778691581109, 7983130477206, 1257688359124, -8167242308412, 629825101852, 5763542313829, -1403630755463, -2871848912941, 1094857977723, 1015613064703, -527354195438, -250382990469, 174510130968, 40234212427, -41084005259, -3181542881, 6933616229, -202049055, -830216501, 92270622, 68449875, -12564205, -3644562, 966733, 106520, -44264, -626, 1120, -50, -12, 1]);
 

\( x^{45} - 12 x^{44} - 50 x^{43} + 1120 x^{42} - 626 x^{41} - 44264 x^{40} + 106520 x^{39} + 966733 x^{38} - 3644562 x^{37} - 12564205 x^{36} + 68449875 x^{35} + 92270622 x^{34} - 830216501 x^{33} - 202049055 x^{32} + 6933616229 x^{31} - 3181542881 x^{30} - 41084005259 x^{29} + 40234212427 x^{28} + 174510130968 x^{27} - 250382990469 x^{26} - 527354195438 x^{25} + 1015613064703 x^{24} + 1094857977723 x^{23} - 2871848912941 x^{22} - 1403630755463 x^{21} + 5763542313829 x^{20} + 629825101852 x^{19} - 8167242308412 x^{18} + 1257688359124 x^{17} + 7983130477206 x^{16} - 2778691581109 x^{15} - 5144196139934 x^{14} + 2589633668016 x^{13} + 2011525458240 x^{12} - 1316532612469 x^{11} - 396963769511 x^{10} + 358975449036 x^{9} + 14178066293 x^{8} - 45168898819 x^{7} + 5258463166 x^{6} + 1537215457 x^{5} - 288043322 x^{4} - 5291206 x^{3} + 2457345 x^{2} - 9309 x - 6089 \)

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:  \(217\!\cdots\!129\)\(\medspace = 7^{30}\cdot 61^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $169.71$
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:  $7, 61$
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:  \(427=7\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{427}(256,·)$, $\chi_{427}(1,·)$, $\chi_{427}(386,·)$, $\chi_{427}(260,·)$, $\chi_{427}(134,·)$, $\chi_{427}(135,·)$, $\chi_{427}(9,·)$, $\chi_{427}(142,·)$, $\chi_{427}(15,·)$, $\chi_{427}(16,·)$, $\chi_{427}(22,·)$, $\chi_{427}(408,·)$, $\chi_{427}(25,·)$, $\chi_{427}(156,·)$, $\chi_{427}(291,·)$, $\chi_{427}(422,·)$, $\chi_{427}(424,·)$, $\chi_{427}(169,·)$, $\chi_{427}(302,·)$, $\chi_{427}(179,·)$, $\chi_{427}(74,·)$, $\chi_{427}(137,·)$, $\chi_{427}(184,·)$, $\chi_{427}(57,·)$, $\chi_{427}(58,·)$, $\chi_{427}(317,·)$, $\chi_{427}(198,·)$, $\chi_{427}(330,·)$, $\chi_{427}(205,·)$, $\chi_{427}(81,·)$, $\chi_{427}(86,·)$, $\chi_{427}(347,·)$, $\chi_{427}(95,·)$, $\chi_{427}(352,·)$, $\chi_{427}(225,·)$, $\chi_{427}(123,·)$, $\chi_{427}(361,·)$, $\chi_{427}(144,·)$, $\chi_{427}(239,·)$, $\chi_{427}(240,·)$, $\chi_{427}(400,·)$, $\chi_{427}(375,·)$, $\chi_{427}(379,·)$, $\chi_{427}(253,·)$, $\chi_{427}(382,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11} a^{27} + \frac{3}{11} a^{26} - \frac{1}{11} a^{25} - \frac{1}{11} a^{24} - \frac{1}{11} a^{23} - \frac{5}{11} a^{22} + \frac{1}{11} a^{21} - \frac{5}{11} a^{20} + \frac{1}{11} a^{18} - \frac{4}{11} a^{17} + \frac{2}{11} a^{16} - \frac{4}{11} a^{15} - \frac{3}{11} a^{14} - \frac{4}{11} a^{13} - \frac{2}{11} a^{12} - \frac{3}{11} a^{11} - \frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{3} + \frac{5}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{28} + \frac{1}{11} a^{26} + \frac{2}{11} a^{25} + \frac{2}{11} a^{24} - \frac{2}{11} a^{23} + \frac{5}{11} a^{22} + \frac{3}{11} a^{21} + \frac{4}{11} a^{20} + \frac{1}{11} a^{19} + \frac{4}{11} a^{18} + \frac{3}{11} a^{17} + \frac{1}{11} a^{16} - \frac{2}{11} a^{15} + \frac{5}{11} a^{14} - \frac{1}{11} a^{13} + \frac{3}{11} a^{12} - \frac{3}{11} a^{11} + \frac{4}{11} a^{10} - \frac{4}{11} a^{8} + \frac{2}{11} a^{7} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{4}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{29} - \frac{1}{11} a^{26} + \frac{3}{11} a^{25} - \frac{1}{11} a^{24} - \frac{5}{11} a^{23} - \frac{3}{11} a^{22} + \frac{3}{11} a^{21} - \frac{5}{11} a^{20} + \frac{4}{11} a^{19} + \frac{2}{11} a^{18} + \frac{5}{11} a^{17} - \frac{4}{11} a^{16} - \frac{2}{11} a^{15} + \frac{2}{11} a^{14} - \frac{4}{11} a^{13} - \frac{1}{11} a^{12} - \frac{4}{11} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} + \frac{5}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{30} - \frac{5}{11} a^{26} - \frac{2}{11} a^{25} + \frac{5}{11} a^{24} - \frac{4}{11} a^{23} - \frac{2}{11} a^{22} - \frac{4}{11} a^{21} - \frac{1}{11} a^{20} + \frac{2}{11} a^{19} - \frac{5}{11} a^{18} + \frac{3}{11} a^{17} - \frac{2}{11} a^{15} + \frac{4}{11} a^{14} - \frac{5}{11} a^{13} + \frac{5}{11} a^{12} - \frac{2}{11} a^{11} + \frac{5}{11} a^{10} + \frac{4}{11} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{5} + \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{31} + \frac{2}{11} a^{26} + \frac{2}{11} a^{24} + \frac{4}{11} a^{23} + \frac{4}{11} a^{22} + \frac{4}{11} a^{21} - \frac{1}{11} a^{20} - \frac{5}{11} a^{19} - \frac{3}{11} a^{18} + \frac{2}{11} a^{17} - \frac{3}{11} a^{16} - \frac{5}{11} a^{15} + \frac{2}{11} a^{14} - \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{1}{11} a^{11} - \frac{5}{11} a^{10} - \frac{2}{11} a^{9} - \frac{3}{11} a^{8} + \frac{3}{11} a^{6} + \frac{4}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{32} + \frac{5}{11} a^{26} + \frac{4}{11} a^{25} - \frac{5}{11} a^{24} - \frac{5}{11} a^{23} + \frac{3}{11} a^{22} - \frac{3}{11} a^{21} + \frac{5}{11} a^{20} - \frac{3}{11} a^{19} + \frac{5}{11} a^{17} + \frac{2}{11} a^{16} - \frac{1}{11} a^{15} + \frac{2}{11} a^{14} - \frac{4}{11} a^{13} + \frac{5}{11} a^{12} + \frac{1}{11} a^{11} - \frac{5}{11} a^{9} + \frac{5}{11} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{3}{11} a^{2} + \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{33} - \frac{3}{11} a^{23} + \frac{3}{11} a^{13} + \frac{4}{11} a^{11} - \frac{1}{11} a^{3} - \frac{4}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{34} - \frac{3}{11} a^{24} + \frac{3}{11} a^{14} + \frac{4}{11} a^{12} - \frac{1}{11} a^{4} - \frac{4}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{11} a^{35} - \frac{3}{11} a^{25} + \frac{3}{11} a^{15} + \frac{4}{11} a^{13} - \frac{1}{11} a^{5} - \frac{4}{11} a^{3} + \frac{4}{11} a^{2}$, $\frac{1}{11} a^{36} - \frac{3}{11} a^{26} + \frac{3}{11} a^{16} + \frac{4}{11} a^{14} - \frac{1}{11} a^{6} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3}$, $\frac{1}{11} a^{37} - \frac{2}{11} a^{26} - \frac{3}{11} a^{25} - \frac{3}{11} a^{24} - \frac{3}{11} a^{23} - \frac{4}{11} a^{22} + \frac{3}{11} a^{21} - \frac{4}{11} a^{20} + \frac{3}{11} a^{18} + \frac{2}{11} a^{17} - \frac{5}{11} a^{16} + \frac{3}{11} a^{15} + \frac{2}{11} a^{14} - \frac{1}{11} a^{13} + \frac{5}{11} a^{12} + \frac{2}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} + \frac{1}{11} a^{3} + \frac{4}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{38} + \frac{3}{11} a^{26} - \frac{5}{11} a^{25} - \frac{5}{11} a^{24} + \frac{5}{11} a^{23} + \frac{4}{11} a^{22} - \frac{2}{11} a^{21} + \frac{1}{11} a^{20} + \frac{3}{11} a^{19} + \frac{4}{11} a^{18} - \frac{2}{11} a^{17} - \frac{4}{11} a^{16} + \frac{5}{11} a^{15} + \frac{4}{11} a^{14} - \frac{3}{11} a^{13} - \frac{2}{11} a^{12} + \frac{2}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{8} - \frac{5}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} + \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{5}{11} a^{2} - \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{39} - \frac{3}{11} a^{26} - \frac{2}{11} a^{25} - \frac{3}{11} a^{24} - \frac{4}{11} a^{23} + \frac{2}{11} a^{22} - \frac{2}{11} a^{21} - \frac{4}{11} a^{20} + \frac{4}{11} a^{19} - \frac{5}{11} a^{18} - \frac{3}{11} a^{17} - \frac{1}{11} a^{16} + \frac{5}{11} a^{15} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{3}{11} a^{12} - \frac{1}{11} a^{11} + \frac{3}{11} a^{10} - \frac{5}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{5}{11} a^{5} + \frac{1}{11} a^{4} + \frac{4}{11} a^{3} + \frac{2}{11} a^{2} - \frac{1}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{40} - \frac{4}{11} a^{26} + \frac{5}{11} a^{25} + \frac{4}{11} a^{24} - \frac{1}{11} a^{23} + \frac{5}{11} a^{22} - \frac{1}{11} a^{21} - \frac{5}{11} a^{19} - \frac{2}{11} a^{17} + \frac{5}{11} a^{15} + \frac{1}{11} a^{14} - \frac{4}{11} a^{13} + \frac{4}{11} a^{12} + \frac{5}{11} a^{11} + \frac{3}{11} a^{10} - \frac{3}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{4}{11} a^{4} + \frac{3}{11} a^{3} + \frac{3}{11} a^{2} + \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{41} - \frac{5}{11} a^{26} - \frac{5}{11} a^{24} + \frac{1}{11} a^{23} + \frac{1}{11} a^{22} + \frac{4}{11} a^{21} - \frac{3}{11} a^{20} + \frac{2}{11} a^{18} - \frac{5}{11} a^{17} + \frac{2}{11} a^{16} - \frac{4}{11} a^{15} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{2}{11} a^{11} - \frac{4}{11} a^{10} + \frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{6} + \frac{1}{11} a^{5} + \frac{3}{11} a^{4} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{72479} a^{42} + \frac{164}{72479} a^{41} - \frac{116}{6589} a^{40} + \frac{1909}{72479} a^{39} + \frac{2341}{72479} a^{38} + \frac{2170}{72479} a^{37} + \frac{1597}{72479} a^{36} - \frac{694}{72479} a^{35} + \frac{1508}{72479} a^{34} + \frac{479}{72479} a^{33} - \frac{537}{72479} a^{32} - \frac{2509}{72479} a^{31} + \frac{730}{72479} a^{30} + \frac{2025}{72479} a^{29} - \frac{18}{6589} a^{28} + \frac{2295}{72479} a^{27} - \frac{3726}{72479} a^{26} + \frac{368}{6589} a^{25} - \frac{25759}{72479} a^{24} + \frac{14412}{72479} a^{23} + \frac{29725}{72479} a^{22} + \frac{14184}{72479} a^{21} + \frac{30047}{72479} a^{20} - \frac{14225}{72479} a^{19} - \frac{13096}{72479} a^{18} - \frac{21892}{72479} a^{17} - \frac{15034}{72479} a^{16} + \frac{9639}{72479} a^{15} - \frac{999}{6589} a^{14} - \frac{25147}{72479} a^{13} - \frac{20939}{72479} a^{12} - \frac{2097}{6589} a^{11} + \frac{2281}{72479} a^{10} - \frac{34447}{72479} a^{9} + \frac{22698}{72479} a^{8} + \frac{24945}{72479} a^{7} - \frac{14909}{72479} a^{6} - \frac{34317}{72479} a^{5} + \frac{340}{6589} a^{4} + \frac{6243}{72479} a^{3} + \frac{1601}{72479} a^{2} - \frac{23213}{72479} a + \frac{9033}{72479}$, $\frac{1}{72479} a^{43} - \frac{1816}{72479} a^{41} + \frac{325}{72479} a^{40} - \frac{1052}{72479} a^{39} + \frac{408}{72479} a^{38} + \frac{1523}{72479} a^{37} + \frac{958}{72479} a^{36} - \frac{298}{6589} a^{35} - \frac{3040}{72479} a^{34} - \frac{25}{72479} a^{33} - \frac{98}{72479} a^{32} - \frac{2901}{72479} a^{31} + \frac{907}{72479} a^{30} - \frac{2848}{72479} a^{29} + \frac{1822}{72479} a^{28} + \frac{2056}{72479} a^{27} - \frac{10843}{72479} a^{26} - \frac{24142}{72479} a^{25} - \frac{4428}{72479} a^{24} - \frac{21104}{72479} a^{23} - \frac{17801}{72479} a^{22} - \frac{287}{6589} a^{21} - \frac{33128}{72479} a^{20} - \frac{25880}{72479} a^{19} + \frac{4194}{72479} a^{18} - \frac{28929}{72479} a^{17} - \frac{28605}{72479} a^{16} - \frac{10425}{72479} a^{15} - \frac{21748}{72479} a^{14} + \frac{17989}{72479} a^{13} - \frac{35118}{72479} a^{12} + \frac{3183}{72479} a^{11} - \frac{13191}{72479} a^{10} - \frac{7712}{72479} a^{9} - \frac{27454}{72479} a^{8} - \frac{20709}{72479} a^{7} + \frac{18952}{72479} a^{6} - \frac{15045}{72479} a^{5} + \frac{25427}{72479} a^{4} - \frac{20723}{72479} a^{3} - \frac{28806}{72479} a^{2} - \frac{32011}{72479} a + \frac{1113}{72479}$, $\frac{1}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{44} - 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\frac{17994527703058728422409278193292752387416076657180697584799351921904866259529244863630667223934539259107846270184495220907252451609276108834540001668354989173342273229462603087595856}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{23} - \frac{11545818294168045123325425309831935874521200222071738553706398230488613047267852406477847942935860150552067281367654028025312775366732647680703812260852456310217362025787983196706158}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{22} - \frac{9764331598738071710735878661139612346589819704448792594610689763987247500495074214055902420082504091269452231940978667848526122577996592648323866637061000665308951781700527094016198}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{21} + \frac{23338025291134970666811783566035255707607223796007836317166846772205300225540645442124604648783264953638324504376363803971236924283793559698089436297410199357795666995863009606082767}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{20} + \frac{18468330489620403790571350416105097897977038029861899113308954336052262837046126104713605268809064466506776129323779894929982245062424705795273255806412927342777320110734508591338031}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{19} - \frac{6403618228183666560641599363782050531669320997802122914442027579859042156537673401606883498465663042490446569047465736343642505609220131315241099250185671020364674069213804096170633}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{18} + \frac{21324889972579739622910536707564558038479143865225506794786454175049104295898537351704699862539648685358848518782766813516967019388757913468086409320576404889804312644364795783891757}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{17} + \frac{32321599187633586194147935088362135063464770067071608018501368740009305490659018251851648250530733239626732434028931698160717661002952244580896424109358547766298478014640523120848942}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{16} + \frac{2327185334086823699134185466039734719174427974027638667266288242429253615755087572162295673119537032642056457650141595988765891351923194445965186689611775303332998267921025935834568}{7319107034276104953613852300826135162859819831225936021455829792156029639950349533671087678297238602589062187198677829438910946486708200794978690007630378395892591459495878463390243} a^{15} + \frac{31425794801674754347184302706565353312789884449279480312287882246340155858550538912156838194012492188453866355380160617614972275279213803913643921737976003147122653638968470052683664}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{14} + \frac{35617847513689202768604441857879437071974676014827797542405957060068695971642243160648797105580284118452040166378300092123816513687995143485731220217879775557103469295219144505832761}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{13} + \frac{752146915732497637076157281741968863157788779585745158987029639554192468690790308140817595973721530950701399859381694011927866152465528424691113981175584345637850299096605933577878}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{12} + \frac{11415073341135731827501900439400499181405680023808996594324459319189953058529713979456743641159411195554725111426715776410565006018199937290738905997718967303368922069602135982027065}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{11} + \frac{12554036513571060741248197158379973378954700059731415094998099474158720085684511024471089060058218927229459422417283352342474931631198537984385612985383231973641343518384760118570116}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{10} + \frac{3550428481396858369853607501729546763882169493163836434908924253282880732503114859473013207702806418491146834328387544454465914467091017507998656621325943614279316168054357787470113}{7319107034276104953613852300826135162859819831225936021455829792156029639950349533671087678297238602589062187198677829438910946486708200794978690007630378395892591459495878463390243} a^{9} + \frac{21380898499886291036204384220017913244913172242492850995867116404298405347577320579121311387782808528844547577151416496749362107178008080321032094095319792171695425502008689009244446}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{8} + \frac{16765554577378952106692464479105132072583442276154178054712013236888714981413356145595562844513401298273525345168079299786874873272392752021647265473346003931560363879040350014195948}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{7} - \frac{19567303903821941409947454067792769864741177723245292841620655530592691139003364096171049056438469335108153092293060286654479768480711450277665560305646609262397081290854204292295785}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{6} - \frac{40024622164554811416436246627226224508754650919163240582702331813080416415002557861701891813933596941558210237212849664521989935520697845443889618248433646037805230458986993346953383}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{5} + \frac{26899459672225155473140675290651811314914139835221766134433420980003720980671842814166262929766074039039292937290944929017294449861590314355808390179374532933656989839345023485595364}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{4} + \frac{16853269521531170832355932655235969972952325684948400039538247141339872353107563969778812524773272655082073441104909894903381212399157941712886326782226267214845357623481614865154045}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{3} + \frac{13089318395317070678113092463041076470619820985363221172323968591444312901111246364296825586991569651102503852841700341444705378420129123054578391722082579626891035450277844859062320}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a^{2} + \frac{40004733438975611041164037748083747532390178476899589499521214126902798505176434871240309559113384810580142627834100033423769486489341415790713626077003517329748492689284994742535746}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673} a - \frac{3893845357831834421752772783243395314969837163453102020957086601224900553711193769695466066031756411823251031760494368532524970067400727898037617854326542368004763933269081474977031}{80510177377037154489752375309087486791458018143485296236014127713716326039453844870381964461269624628479684059185456123828020411353790208744765590083934162354818506054454663097292673}$

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

$C_3\times C_{15}$ (as 45T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.3721.1, 3.3.182329.2, 3.3.182329.1, 5.5.13845841.1, 9.9.6061320523197289.1, 15.15.749787887447605250170677896929.1, 15.15.9876832533361318095112441.1, 15.15.2789960729192539135885092454472809.2, 15.15.2789960729192539135885092454472809.1

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 $15^{3}$ $15^{3}$ $15^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{15}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{9}$ $15^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$

In the table, R denotes a ramified prime. 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
7Data not computed
61Data not computed