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Properties

Label 32.0.37648753674...2689.1
Degree $32$
Signature $[0, 16]$
Discriminant $11^{16}\cdot 17^{30}$
Root discriminant $47.23$
Ramified primes $11, 17$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, -14348907, -9565938, 7971615, 531441, -2834352, 767637, 688905, -485514, -67797, 184437, -38880, -48519, 29133, 6462, -11865, 1801, -3955, 718, 1079, -599, -160, 253, -31, -74, 35, 13, -16, 1, 5, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 2*x^30 + 5*x^29 + x^28 - 16*x^27 + 13*x^26 + 35*x^25 - 74*x^24 - 31*x^23 + 253*x^22 - 160*x^21 - 599*x^20 + 1079*x^19 + 718*x^18 - 3955*x^17 + 1801*x^16 - 11865*x^15 + 6462*x^14 + 29133*x^13 - 48519*x^12 - 38880*x^11 + 184437*x^10 - 67797*x^9 - 485514*x^8 + 688905*x^7 + 767637*x^6 - 2834352*x^5 + 531441*x^4 + 7971615*x^3 - 9565938*x^2 - 14348907*x + 43046721)
 
gp: K = bnfinit(x^32 - x^31 - 2*x^30 + 5*x^29 + x^28 - 16*x^27 + 13*x^26 + 35*x^25 - 74*x^24 - 31*x^23 + 253*x^22 - 160*x^21 - 599*x^20 + 1079*x^19 + 718*x^18 - 3955*x^17 + 1801*x^16 - 11865*x^15 + 6462*x^14 + 29133*x^13 - 48519*x^12 - 38880*x^11 + 184437*x^10 - 67797*x^9 - 485514*x^8 + 688905*x^7 + 767637*x^6 - 2834352*x^5 + 531441*x^4 + 7971615*x^3 - 9565938*x^2 - 14348907*x + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 2 x^{30} + 5 x^{29} + x^{28} - 16 x^{27} + 13 x^{26} + 35 x^{25} - 74 x^{24} - 31 x^{23} + 253 x^{22} - 160 x^{21} - 599 x^{20} + 1079 x^{19} + 718 x^{18} - 3955 x^{17} + 1801 x^{16} - 11865 x^{15} + 6462 x^{14} + 29133 x^{13} - 48519 x^{12} - 38880 x^{11} + 184437 x^{10} - 67797 x^{9} - 485514 x^{8} + 688905 x^{7} + 767637 x^{6} - 2834352 x^{5} + 531441 x^{4} + 7971615 x^{3} - 9565938 x^{2} - 14348907 x + 43046721 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(376487536747625684443481098520430035050561572415862689=11^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.23$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(131,·)$, $\chi_{187}(133,·)$, $\chi_{187}(10,·)$, $\chi_{187}(12,·)$, $\chi_{187}(142,·)$, $\chi_{187}(144,·)$, $\chi_{187}(21,·)$, $\chi_{187}(23,·)$, $\chi_{187}(155,·)$, $\chi_{187}(32,·)$, $\chi_{187}(164,·)$, $\chi_{187}(166,·)$, $\chi_{187}(43,·)$, $\chi_{187}(45,·)$, $\chi_{187}(175,·)$, $\chi_{187}(177,·)$, $\chi_{187}(54,·)$, $\chi_{187}(56,·)$, $\chi_{187}(186,·)$, $\chi_{187}(65,·)$, $\chi_{187}(67,·)$, $\chi_{187}(76,·)$, $\chi_{187}(78,·)$, $\chi_{187}(87,·)$, $\chi_{187}(89,·)$, $\chi_{187}(98,·)$, $\chi_{187}(100,·)$, $\chi_{187}(109,·)$, $\chi_{187}(111,·)$, $\chi_{187}(120,·)$, $\chi_{187}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5403} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{353}{1801}$, $\frac{1}{16209} a^{18} - \frac{1}{16209} a^{17} - \frac{2}{9} a^{16} - \frac{4}{9} a^{15} + \frac{1}{9} a^{14} + \frac{2}{9} a^{13} + \frac{4}{9} a^{12} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} - \frac{4}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1448}{5403} a + \frac{718}{1801}$, $\frac{1}{48627} a^{19} - \frac{1}{48627} a^{18} - \frac{2}{48627} a^{17} - \frac{4}{27} a^{16} + \frac{10}{27} a^{15} + \frac{2}{27} a^{14} - \frac{5}{27} a^{13} - \frac{1}{27} a^{12} - \frac{11}{27} a^{11} - \frac{13}{27} a^{10} - \frac{8}{27} a^{9} - \frac{7}{27} a^{8} + \frac{4}{27} a^{7} - \frac{10}{27} a^{6} - \frac{2}{27} a^{5} + \frac{5}{27} a^{4} + \frac{1}{27} a^{3} - \frac{3955}{16209} a^{2} + \frac{718}{5403} a - \frac{722}{1801}$, $\frac{1}{145881} a^{20} - \frac{1}{145881} a^{19} - \frac{2}{145881} a^{18} + \frac{5}{145881} a^{17} - \frac{17}{81} a^{16} + \frac{29}{81} a^{15} + \frac{22}{81} a^{14} - \frac{28}{81} a^{13} - \frac{38}{81} a^{12} - \frac{40}{81} a^{11} - \frac{8}{81} a^{10} - \frac{34}{81} a^{9} - \frac{23}{81} a^{8} - \frac{37}{81} a^{7} + \frac{25}{81} a^{6} + \frac{5}{81} a^{5} + \frac{1}{81} a^{4} - \frac{3955}{48627} a^{3} + \frac{718}{16209} a^{2} + \frac{1079}{5403} a - \frac{599}{1801}$, $\frac{1}{437643} a^{21} - \frac{1}{437643} a^{20} - \frac{2}{437643} a^{19} + \frac{5}{437643} a^{18} + \frac{1}{437643} a^{17} + \frac{29}{243} a^{16} + \frac{22}{243} a^{15} - \frac{109}{243} a^{14} + \frac{43}{243} a^{13} + \frac{41}{243} a^{12} + \frac{73}{243} a^{11} + \frac{47}{243} a^{10} - \frac{23}{243} a^{9} - \frac{118}{243} a^{8} - \frac{56}{243} a^{7} - \frac{76}{243} a^{6} + \frac{1}{243} a^{5} - \frac{3955}{145881} a^{4} + \frac{718}{48627} a^{3} + \frac{1079}{16209} a^{2} - \frac{599}{5403} a - \frac{160}{1801}$, $\frac{1}{1312929} a^{22} - \frac{1}{1312929} a^{21} - \frac{2}{1312929} a^{20} + \frac{5}{1312929} a^{19} + \frac{1}{1312929} a^{18} - \frac{16}{1312929} a^{17} - \frac{221}{729} a^{16} + \frac{134}{729} a^{15} - \frac{200}{729} a^{14} - \frac{202}{729} a^{13} + \frac{73}{729} a^{12} - \frac{196}{729} a^{11} - \frac{23}{729} a^{10} - \frac{118}{729} a^{9} + \frac{187}{729} a^{8} + \frac{167}{729} a^{7} + \frac{1}{729} a^{6} - \frac{3955}{437643} a^{5} + \frac{718}{145881} a^{4} + \frac{1079}{48627} a^{3} - \frac{599}{16209} a^{2} - \frac{160}{5403} a + \frac{253}{1801}$, $\frac{1}{3938787} a^{23} - \frac{1}{3938787} a^{22} - \frac{2}{3938787} a^{21} + \frac{5}{3938787} a^{20} + \frac{1}{3938787} a^{19} - \frac{16}{3938787} a^{18} + \frac{13}{3938787} a^{17} - \frac{595}{2187} a^{16} - \frac{929}{2187} a^{15} + \frac{527}{2187} a^{14} + \frac{73}{2187} a^{13} + \frac{533}{2187} a^{12} - \frac{752}{2187} a^{11} - \frac{847}{2187} a^{10} + \frac{916}{2187} a^{9} - \frac{562}{2187} a^{8} + \frac{1}{2187} a^{7} - \frac{3955}{1312929} a^{6} + \frac{718}{437643} a^{5} + \frac{1079}{145881} a^{4} - \frac{599}{48627} a^{3} - \frac{160}{16209} a^{2} + \frac{253}{5403} a - \frac{31}{1801}$, $\frac{1}{11816361} a^{24} - \frac{1}{11816361} a^{23} - \frac{2}{11816361} a^{22} + \frac{5}{11816361} a^{21} + \frac{1}{11816361} a^{20} - \frac{16}{11816361} a^{19} + \frac{13}{11816361} a^{18} + \frac{35}{11816361} a^{17} - \frac{929}{6561} a^{16} + \frac{2714}{6561} a^{15} + \frac{73}{6561} a^{14} - \frac{1654}{6561} a^{13} + \frac{1435}{6561} a^{12} - \frac{3034}{6561} a^{11} - \frac{1271}{6561} a^{10} - \frac{2749}{6561} a^{9} + \frac{1}{6561} a^{8} - \frac{3955}{3938787} a^{7} + \frac{718}{1312929} a^{6} + \frac{1079}{437643} a^{5} - \frac{599}{145881} a^{4} - \frac{160}{48627} a^{3} + \frac{253}{16209} a^{2} - \frac{31}{5403} a - \frac{74}{1801}$, $\frac{1}{35449083} a^{25} - \frac{1}{35449083} a^{24} - \frac{2}{35449083} a^{23} + \frac{5}{35449083} a^{22} + \frac{1}{35449083} a^{21} - \frac{16}{35449083} a^{20} + \frac{13}{35449083} a^{19} + \frac{35}{35449083} a^{18} - \frac{74}{35449083} a^{17} - \frac{3847}{19683} a^{16} + \frac{6634}{19683} a^{15} + \frac{4907}{19683} a^{14} - \frac{5126}{19683} a^{13} - \frac{9595}{19683} a^{12} + \frac{5290}{19683} a^{11} + \frac{3812}{19683} a^{10} + \frac{1}{19683} a^{9} - \frac{3955}{11816361} a^{8} + \frac{718}{3938787} a^{7} + \frac{1079}{1312929} a^{6} - \frac{599}{437643} a^{5} - \frac{160}{145881} a^{4} + \frac{253}{48627} a^{3} - \frac{31}{16209} a^{2} - \frac{74}{5403} a + \frac{35}{1801}$, $\frac{1}{106347249} a^{26} - \frac{1}{106347249} a^{25} - \frac{2}{106347249} a^{24} + \frac{5}{106347249} a^{23} + \frac{1}{106347249} a^{22} - \frac{16}{106347249} a^{21} + \frac{13}{106347249} a^{20} + \frac{35}{106347249} a^{19} - \frac{74}{106347249} a^{18} - \frac{31}{106347249} a^{17} - \frac{13049}{59049} a^{16} + \frac{24590}{59049} a^{15} + \frac{14557}{59049} a^{14} - \frac{29278}{59049} a^{13} - \frac{14393}{59049} a^{12} - \frac{15871}{59049} a^{11} + \frac{1}{59049} a^{10} - \frac{3955}{35449083} a^{9} + \frac{718}{11816361} a^{8} + \frac{1079}{3938787} a^{7} - \frac{599}{1312929} a^{6} - \frac{160}{437643} a^{5} + \frac{253}{145881} a^{4} - \frac{31}{48627} a^{3} - \frac{74}{16209} a^{2} + \frac{35}{5403} a + \frac{13}{1801}$, $\frac{1}{319041747} a^{27} - \frac{1}{319041747} a^{26} - \frac{2}{319041747} a^{25} + \frac{5}{319041747} a^{24} + \frac{1}{319041747} a^{23} - \frac{16}{319041747} a^{22} + \frac{13}{319041747} a^{21} + \frac{35}{319041747} a^{20} - \frac{74}{319041747} a^{19} - \frac{31}{319041747} a^{18} + \frac{253}{319041747} a^{17} + \frac{24590}{177147} a^{16} + \frac{14557}{177147} a^{15} - \frac{88327}{177147} a^{14} + \frac{44656}{177147} a^{13} + \frac{43178}{177147} a^{12} + \frac{1}{177147} a^{11} - \frac{3955}{106347249} a^{10} + \frac{718}{35449083} a^{9} + \frac{1079}{11816361} a^{8} - \frac{599}{3938787} a^{7} - \frac{160}{1312929} a^{6} + \frac{253}{437643} a^{5} - \frac{31}{145881} a^{4} - \frac{74}{48627} a^{3} + \frac{35}{16209} a^{2} + \frac{13}{5403} a - \frac{16}{1801}$, $\frac{1}{957125241} a^{28} - \frac{1}{957125241} a^{27} - \frac{2}{957125241} a^{26} + \frac{5}{957125241} a^{25} + \frac{1}{957125241} a^{24} - \frac{16}{957125241} a^{23} + \frac{13}{957125241} a^{22} + \frac{35}{957125241} a^{21} - \frac{74}{957125241} a^{20} - \frac{31}{957125241} a^{19} + \frac{253}{957125241} a^{18} - \frac{160}{957125241} a^{17} - \frac{162590}{531441} a^{16} + \frac{88820}{531441} a^{15} - \frac{132491}{531441} a^{14} - \frac{133969}{531441} a^{13} + \frac{1}{531441} a^{12} - \frac{3955}{319041747} a^{11} + \frac{718}{106347249} a^{10} + \frac{1079}{35449083} a^{9} - \frac{599}{11816361} a^{8} - \frac{160}{3938787} a^{7} + \frac{253}{1312929} a^{6} - \frac{31}{437643} a^{5} - \frac{74}{145881} a^{4} + \frac{35}{48627} a^{3} + \frac{13}{16209} a^{2} - \frac{16}{5403} a + \frac{1}{1801}$, $\frac{1}{2871375723} a^{29} - \frac{1}{2871375723} a^{28} - \frac{2}{2871375723} a^{27} + \frac{5}{2871375723} a^{26} + \frac{1}{2871375723} a^{25} - \frac{16}{2871375723} a^{24} + \frac{13}{2871375723} a^{23} + \frac{35}{2871375723} a^{22} - \frac{74}{2871375723} a^{21} - \frac{31}{2871375723} a^{20} + \frac{253}{2871375723} a^{19} - \frac{160}{2871375723} a^{18} - \frac{599}{2871375723} a^{17} - \frac{442621}{1594323} a^{16} - \frac{663932}{1594323} a^{15} + \frac{397472}{1594323} a^{14} + \frac{1}{1594323} a^{13} - \frac{3955}{957125241} a^{12} + \frac{718}{319041747} a^{11} + \frac{1079}{106347249} a^{10} - \frac{599}{35449083} a^{9} - \frac{160}{11816361} a^{8} + \frac{253}{3938787} a^{7} - \frac{31}{1312929} a^{6} - \frac{74}{437643} a^{5} + \frac{35}{145881} a^{4} + \frac{13}{48627} a^{3} - \frac{16}{16209} a^{2} + \frac{1}{5403} a + \frac{5}{1801}$, $\frac{1}{8614127169} a^{30} - \frac{1}{8614127169} a^{29} - \frac{2}{8614127169} a^{28} + \frac{5}{8614127169} a^{27} + \frac{1}{8614127169} a^{26} - \frac{16}{8614127169} a^{25} + \frac{13}{8614127169} a^{24} + \frac{35}{8614127169} a^{23} - \frac{74}{8614127169} a^{22} - \frac{31}{8614127169} a^{21} + \frac{253}{8614127169} a^{20} - \frac{160}{8614127169} a^{19} - \frac{599}{8614127169} a^{18} + \frac{1079}{8614127169} a^{17} - \frac{663932}{4782969} a^{16} + \frac{1991795}{4782969} a^{15} + \frac{1}{4782969} a^{14} - \frac{3955}{2871375723} a^{13} + \frac{718}{957125241} a^{12} + \frac{1079}{319041747} a^{11} - \frac{599}{106347249} a^{10} - \frac{160}{35449083} a^{9} + \frac{253}{11816361} a^{8} - \frac{31}{3938787} a^{7} - \frac{74}{1312929} a^{6} + \frac{35}{437643} a^{5} + \frac{13}{145881} a^{4} - \frac{16}{48627} a^{3} + \frac{1}{16209} a^{2} + \frac{5}{5403} a - \frac{2}{1801}$, $\frac{1}{25842381507} a^{31} - \frac{1}{25842381507} a^{30} - \frac{2}{25842381507} a^{29} + \frac{5}{25842381507} a^{28} + \frac{1}{25842381507} a^{27} - \frac{16}{25842381507} a^{26} + \frac{13}{25842381507} a^{25} + \frac{35}{25842381507} a^{24} - \frac{74}{25842381507} a^{23} - \frac{31}{25842381507} a^{22} + \frac{253}{25842381507} a^{21} - \frac{160}{25842381507} a^{20} - \frac{599}{25842381507} a^{19} + \frac{1079}{25842381507} a^{18} + \frac{718}{25842381507} a^{17} + \frac{1991795}{14348907} a^{16} + \frac{1}{14348907} a^{15} - \frac{3955}{8614127169} a^{14} + \frac{718}{2871375723} a^{13} + \frac{1079}{957125241} a^{12} - \frac{599}{319041747} a^{11} - \frac{160}{106347249} a^{10} + \frac{253}{35449083} a^{9} - \frac{31}{11816361} a^{8} - \frac{74}{3938787} a^{7} + \frac{35}{1312929} a^{6} + \frac{13}{437643} a^{5} - \frac{16}{145881} a^{4} + \frac{1}{48627} a^{3} + \frac{5}{16209} a^{2} - \frac{2}{5403} a - \frac{1}{1801}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{599}{2871375723} a^{30} + \frac{5163856}{2871375723} a^{13} \) (order $34$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{-11}, \sqrt{17})\), 4.4.4913.1, 4.0.594473.1, 8.0.353398147729.1, \(\Q(\zeta_{17})^+\), 8.0.6007768511393.1, 16.0.36093282486485263170800449.1, \(\Q(\zeta_{17})\), 16.16.613585802270249473903607633.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
17Data not computed