LMFDB - Number field 32.0.376487536747625684443481098520430035050561572415862689.1

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Normalized defining polynomial

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(y^32 - y^31 - 2*y^30 + 5*y^29 + y^28 - 16*y^27 + 13*y^26 + 35*y^25 - 74*y^24 - 31*y^23 + 253*y^22 - 160*y^21 - 599*y^20 + 1079*y^19 + 718*y^18 - 3955*y^17 + 1801*y^16 - 11865*y^15 + 6462*y^14 + 29133*y^13 - 48519*y^12 - 38880*y^11 + 184437*y^10 - 67797*y^9 - 485514*y^8 + 688905*y^7 + 767637*y^6 - 2834352*y^5 + 531441*y^4 + 7971615*y^3 - 9565938*y^2 - 14348907*y + 43046721, 1)

magma: R<x> := PolynomialRing(Rationals()); 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);

oscar: Qx, x = PolynomialRing(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)

$ x^{32} - x^{31} - 2 x^{30} + 5 x^{29} + x^{28} - 16 x^{27} + 13 x^{26} + 35 x^{25} - 74 x^{24} + \cdots + 43046721 $ 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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$376487536747625684443481098520430035050561572415862689$ 376487536747625684443481098520430035050561572415862689 $\medspace = 11^{16}\cdot 17^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$47.23$	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:	$11^{1/2}17^{15/16}\approx 47.232638461641514$
Ramified primes:	$11$, $17$ 11, 17	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$32$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
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.
Reflex fields:	unavailable$^{32768}$

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

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{599}{2871375723} a^{30} + \frac{5163856}{2871375723} a^{13} $ -(599)/(2871375723)a^(30) + (5163856)/(2871375723)a^(13) (order $34$)	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^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)

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^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);

[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^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);

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^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);

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

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

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)

An abelian group of order 32

The 32 conjugacy class representatives for $C_2\times C_{16}$

Character table for $C_2\times C_{16}$

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.

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

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

# 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$11$ 11		Deg $32$	$2$	$16$	$16$
$17$ 17		Deg $32$	$16$	$2$	$30$

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