LMFDB - Number field 36.0.166990115557548038315544519372094023948173088869511853538488801.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489)

gp: K = bnfinit(y^36 - y^35 - 2*y^34 + 5*y^33 + y^32 - 16*y^31 + 13*y^30 + 35*y^29 - 74*y^28 - 31*y^27 + 253*y^26 - 160*y^25 - 599*y^24 + 1079*y^23 + 718*y^22 - 3955*y^21 + 1801*y^20 + 10064*y^19 - 15467*y^18 + 30192*y^17 + 16209*y^16 - 106785*y^15 + 58158*y^14 + 262197*y^13 - 436671*y^12 - 349920*y^11 + 1659933*y^10 - 610173*y^9 - 4369626*y^8 + 6200145*y^7 + 6908733*y^6 - 25509168*y^5 + 4782969*y^4 + 71744535*y^3 - 86093442*y^2 - 129140163*y + 387420489, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489)

$ x^{36} - x^{35} - 2 x^{34} + 5 x^{33} + x^{32} - 16 x^{31} + 13 x^{30} + 35 x^{29} - 74 x^{28} + \cdots + 387420489 $ x^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$36$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 18]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$166990115557548038315544519372094023948173088869511853538488801$ 166990115557548038315544519372094023948173088869511853538488801 $\medspace = 11^{18}\cdot 19^{34}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$53.51$	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:	$11^{1/2}19^{17/18}\approx 53.50668890125934$
Ramified primes:	$11$, $19$ 11, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$36$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$209=11\cdot 19$
Dirichlet character group:	$\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(131,·)$, $\chi_{209}(10,·)$, $\chi_{209}(12,·)$, $\chi_{209}(142,·)$, $\chi_{209}(144,·)$, $\chi_{209}(21,·)$, $\chi_{209}(23,·)$, $\chi_{209}(153,·)$, $\chi_{209}(155,·)$, $\chi_{209}(32,·)$, $\chi_{209}(34,·)$, $\chi_{209}(164,·)$, $\chi_{209}(166,·)$, $\chi_{209}(43,·)$, $\chi_{209}(45,·)$, $\chi_{209}(175,·)$, $\chi_{209}(177,·)$, $\chi_{209}(54,·)$, $\chi_{209}(56,·)$, $\chi_{209}(186,·)$, $\chi_{209}(188,·)$, $\chi_{209}(65,·)$, $\chi_{209}(67,·)$, $\chi_{209}(197,·)$, $\chi_{209}(199,·)$, $\chi_{209}(78,·)$, $\chi_{209}(208,·)$, $\chi_{209}(87,·)$, $\chi_{209}(89,·)$, $\chi_{209}(98,·)$, $\chi_{209}(100,·)$, $\chi_{209}(109,·)$, $\chi_{209}(111,·)$, $\chi_{209}(120,·)$, $\chi_{209}(122,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{46401}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{5403}{15467}$, $\frac{1}{139203}a^{20}-\frac{1}{139203}a^{19}-\frac{4}{9}a^{18}+\frac{1}{9}a^{17}+\frac{2}{9}a^{16}+\frac{4}{9}a^{15}-\frac{1}{9}a^{14}-\frac{2}{9}a^{13}-\frac{4}{9}a^{12}+\frac{1}{9}a^{11}+\frac{2}{9}a^{10}+\frac{4}{9}a^{9}-\frac{1}{9}a^{8}-\frac{2}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{2}{9}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}+\frac{10064}{46401}a+\frac{1801}{15467}$, $\frac{1}{417609}a^{21}-\frac{1}{417609}a^{20}-\frac{2}{417609}a^{19}-\frac{8}{27}a^{18}-\frac{7}{27}a^{17}+\frac{4}{27}a^{16}-\frac{10}{27}a^{15}-\frac{2}{27}a^{14}+\frac{5}{27}a^{13}+\frac{1}{27}a^{12}+\frac{11}{27}a^{11}+\frac{13}{27}a^{10}+\frac{8}{27}a^{9}+\frac{7}{27}a^{8}-\frac{4}{27}a^{7}+\frac{10}{27}a^{6}+\frac{2}{27}a^{5}-\frac{5}{27}a^{4}-\frac{1}{27}a^{3}+\frac{10064}{139203}a^{2}+\frac{1801}{46401}a-\frac{3955}{15467}$, $\frac{1}{1252827}a^{22}-\frac{1}{1252827}a^{21}-\frac{2}{1252827}a^{20}+\frac{5}{1252827}a^{19}+\frac{20}{81}a^{18}+\frac{4}{81}a^{17}+\frac{17}{81}a^{16}-\frac{29}{81}a^{15}-\frac{22}{81}a^{14}+\frac{28}{81}a^{13}+\frac{38}{81}a^{12}+\frac{40}{81}a^{11}+\frac{8}{81}a^{10}+\frac{34}{81}a^{9}+\frac{23}{81}a^{8}+\frac{37}{81}a^{7}-\frac{25}{81}a^{6}-\frac{5}{81}a^{5}-\frac{1}{81}a^{4}+\frac{10064}{417609}a^{3}+\frac{1801}{139203}a^{2}-\frac{3955}{46401}a+\frac{718}{15467}$, $\frac{1}{3758481}a^{23}-\frac{1}{3758481}a^{22}-\frac{2}{3758481}a^{21}+\frac{5}{3758481}a^{20}+\frac{1}{3758481}a^{19}-\frac{77}{243}a^{18}+\frac{17}{243}a^{17}-\frac{29}{243}a^{16}-\frac{22}{243}a^{15}+\frac{109}{243}a^{14}-\frac{43}{243}a^{13}-\frac{41}{243}a^{12}-\frac{73}{243}a^{11}-\frac{47}{243}a^{10}+\frac{23}{243}a^{9}+\frac{118}{243}a^{8}+\frac{56}{243}a^{7}+\frac{76}{243}a^{6}-\frac{1}{243}a^{5}+\frac{10064}{1252827}a^{4}+\frac{1801}{417609}a^{3}-\frac{3955}{139203}a^{2}+\frac{718}{46401}a+\frac{1079}{15467}$, $\frac{1}{11275443}a^{24}-\frac{1}{11275443}a^{23}-\frac{2}{11275443}a^{22}+\frac{5}{11275443}a^{21}+\frac{1}{11275443}a^{20}-\frac{16}{11275443}a^{19}-\frac{226}{729}a^{18}-\frac{272}{729}a^{17}+\frac{221}{729}a^{16}-\frac{134}{729}a^{15}+\frac{200}{729}a^{14}+\frac{202}{729}a^{13}-\frac{73}{729}a^{12}+\frac{196}{729}a^{11}+\frac{23}{729}a^{10}+\frac{118}{729}a^{9}-\frac{187}{729}a^{8}-\frac{167}{729}a^{7}-\frac{1}{729}a^{6}+\frac{10064}{3758481}a^{5}+\frac{1801}{1252827}a^{4}-\frac{3955}{417609}a^{3}+\frac{718}{139203}a^{2}+\frac{1079}{46401}a-\frac{599}{15467}$, $\frac{1}{33826329}a^{25}-\frac{1}{33826329}a^{24}-\frac{2}{33826329}a^{23}+\frac{5}{33826329}a^{22}+\frac{1}{33826329}a^{21}-\frac{16}{33826329}a^{20}+\frac{13}{33826329}a^{19}+\frac{457}{2187}a^{18}+\frac{221}{2187}a^{17}+\frac{595}{2187}a^{16}+\frac{929}{2187}a^{15}-\frac{527}{2187}a^{14}-\frac{73}{2187}a^{13}-\frac{533}{2187}a^{12}+\frac{752}{2187}a^{11}+\frac{847}{2187}a^{10}-\frac{916}{2187}a^{9}+\frac{562}{2187}a^{8}-\frac{1}{2187}a^{7}+\frac{10064}{11275443}a^{6}+\frac{1801}{3758481}a^{5}-\frac{3955}{1252827}a^{4}+\frac{718}{417609}a^{3}+\frac{1079}{139203}a^{2}-\frac{599}{46401}a-\frac{160}{15467}$, $\frac{1}{101478987}a^{26}-\frac{1}{101478987}a^{25}-\frac{2}{101478987}a^{24}+\frac{5}{101478987}a^{23}+\frac{1}{101478987}a^{22}-\frac{16}{101478987}a^{21}+\frac{13}{101478987}a^{20}+\frac{35}{101478987}a^{19}+\frac{221}{6561}a^{18}-\frac{1592}{6561}a^{17}+\frac{929}{6561}a^{16}-\frac{2714}{6561}a^{15}-\frac{73}{6561}a^{14}+\frac{1654}{6561}a^{13}-\frac{1435}{6561}a^{12}+\frac{3034}{6561}a^{11}+\frac{1271}{6561}a^{10}+\frac{2749}{6561}a^{9}-\frac{1}{6561}a^{8}+\frac{10064}{33826329}a^{7}+\frac{1801}{11275443}a^{6}-\frac{3955}{3758481}a^{5}+\frac{718}{1252827}a^{4}+\frac{1079}{417609}a^{3}-\frac{599}{139203}a^{2}-\frac{160}{46401}a+\frac{253}{15467}$, $\frac{1}{304436961}a^{27}-\frac{1}{304436961}a^{26}-\frac{2}{304436961}a^{25}+\frac{5}{304436961}a^{24}+\frac{1}{304436961}a^{23}-\frac{16}{304436961}a^{22}+\frac{13}{304436961}a^{21}+\frac{35}{304436961}a^{20}-\frac{74}{304436961}a^{19}-\frac{1592}{19683}a^{18}+\frac{929}{19683}a^{17}+\frac{3847}{19683}a^{16}-\frac{6634}{19683}a^{15}-\frac{4907}{19683}a^{14}+\frac{5126}{19683}a^{13}+\frac{9595}{19683}a^{12}-\frac{5290}{19683}a^{11}-\frac{3812}{19683}a^{10}-\frac{1}{19683}a^{9}+\frac{10064}{101478987}a^{8}+\frac{1801}{33826329}a^{7}-\frac{3955}{11275443}a^{6}+\frac{718}{3758481}a^{5}+\frac{1079}{1252827}a^{4}-\frac{599}{417609}a^{3}-\frac{160}{139203}a^{2}+\frac{253}{46401}a-\frac{31}{15467}$, $\frac{1}{913310883}a^{28}-\frac{1}{913310883}a^{27}-\frac{2}{913310883}a^{26}+\frac{5}{913310883}a^{25}+\frac{1}{913310883}a^{24}-\frac{16}{913310883}a^{23}+\frac{13}{913310883}a^{22}+\frac{35}{913310883}a^{21}-\frac{74}{913310883}a^{20}-\frac{31}{913310883}a^{19}+\frac{20612}{59049}a^{18}-\frac{15836}{59049}a^{17}+\frac{13049}{59049}a^{16}-\frac{24590}{59049}a^{15}-\frac{14557}{59049}a^{14}+\frac{29278}{59049}a^{13}+\frac{14393}{59049}a^{12}+\frac{15871}{59049}a^{11}-\frac{1}{59049}a^{10}+\frac{10064}{304436961}a^{9}+\frac{1801}{101478987}a^{8}-\frac{3955}{33826329}a^{7}+\frac{718}{11275443}a^{6}+\frac{1079}{3758481}a^{5}-\frac{599}{1252827}a^{4}-\frac{160}{417609}a^{3}+\frac{253}{139203}a^{2}-\frac{31}{46401}a-\frac{74}{15467}$, $\frac{1}{2739932649}a^{29}-\frac{1}{2739932649}a^{28}-\frac{2}{2739932649}a^{27}+\frac{5}{2739932649}a^{26}+\frac{1}{2739932649}a^{25}-\frac{16}{2739932649}a^{24}+\frac{13}{2739932649}a^{23}+\frac{35}{2739932649}a^{22}-\frac{74}{2739932649}a^{21}-\frac{31}{2739932649}a^{20}+\frac{253}{2739932649}a^{19}+\frac{43213}{177147}a^{18}+\frac{72098}{177147}a^{17}-\frac{24590}{177147}a^{16}-\frac{14557}{177147}a^{15}+\frac{88327}{177147}a^{14}-\frac{44656}{177147}a^{13}-\frac{43178}{177147}a^{12}-\frac{1}{177147}a^{11}+\frac{10064}{913310883}a^{10}+\frac{1801}{304436961}a^{9}-\frac{3955}{101478987}a^{8}+\frac{718}{33826329}a^{7}+\frac{1079}{11275443}a^{6}-\frac{599}{3758481}a^{5}-\frac{160}{1252827}a^{4}+\frac{253}{417609}a^{3}-\frac{31}{139203}a^{2}-\frac{74}{46401}a+\frac{35}{15467}$, $\frac{1}{8219797947}a^{30}-\frac{1}{8219797947}a^{29}-\frac{2}{8219797947}a^{28}+\frac{5}{8219797947}a^{27}+\frac{1}{8219797947}a^{26}-\frac{16}{8219797947}a^{25}+\frac{13}{8219797947}a^{24}+\frac{35}{8219797947}a^{23}-\frac{74}{8219797947}a^{22}-\frac{31}{8219797947}a^{21}+\frac{253}{8219797947}a^{20}-\frac{160}{8219797947}a^{19}+\frac{249245}{531441}a^{18}+\frac{152557}{531441}a^{17}+\frac{162590}{531441}a^{16}-\frac{88820}{531441}a^{15}+\frac{132491}{531441}a^{14}+\frac{133969}{531441}a^{13}-\frac{1}{531441}a^{12}+\frac{10064}{2739932649}a^{11}+\frac{1801}{913310883}a^{10}-\frac{3955}{304436961}a^{9}+\frac{718}{101478987}a^{8}+\frac{1079}{33826329}a^{7}-\frac{599}{11275443}a^{6}-\frac{160}{3758481}a^{5}+\frac{253}{1252827}a^{4}-\frac{31}{417609}a^{3}-\frac{74}{139203}a^{2}+\frac{35}{46401}a+\frac{13}{15467}$, $\frac{1}{24659393841}a^{31}-\frac{1}{24659393841}a^{30}-\frac{2}{24659393841}a^{29}+\frac{5}{24659393841}a^{28}+\frac{1}{24659393841}a^{27}-\frac{16}{24659393841}a^{26}+\frac{13}{24659393841}a^{25}+\frac{35}{24659393841}a^{24}-\frac{74}{24659393841}a^{23}-\frac{31}{24659393841}a^{22}+\frac{253}{24659393841}a^{21}-\frac{160}{24659393841}a^{20}-\frac{599}{24659393841}a^{19}+\frac{152557}{1594323}a^{18}+\frac{694031}{1594323}a^{17}+\frac{442621}{1594323}a^{16}+\frac{663932}{1594323}a^{15}-\frac{397472}{1594323}a^{14}-\frac{1}{1594323}a^{13}+\frac{10064}{8219797947}a^{12}+\frac{1801}{2739932649}a^{11}-\frac{3955}{913310883}a^{10}+\frac{718}{304436961}a^{9}+\frac{1079}{101478987}a^{8}-\frac{599}{33826329}a^{7}-\frac{160}{11275443}a^{6}+\frac{253}{3758481}a^{5}-\frac{31}{1252827}a^{4}-\frac{74}{417609}a^{3}+\frac{35}{139203}a^{2}+\frac{13}{46401}a-\frac{16}{15467}$, $\frac{1}{73978181523}a^{32}-\frac{1}{73978181523}a^{31}-\frac{2}{73978181523}a^{30}+\frac{5}{73978181523}a^{29}+\frac{1}{73978181523}a^{28}-\frac{16}{73978181523}a^{27}+\frac{13}{73978181523}a^{26}+\frac{35}{73978181523}a^{25}-\frac{74}{73978181523}a^{24}-\frac{31}{73978181523}a^{23}+\frac{253}{73978181523}a^{22}-\frac{160}{73978181523}a^{21}-\frac{599}{73978181523}a^{20}+\frac{1079}{73978181523}a^{19}+\frac{2288354}{4782969}a^{18}+\frac{2036944}{4782969}a^{17}+\frac{663932}{4782969}a^{16}-\frac{1991795}{4782969}a^{15}-\frac{1}{4782969}a^{14}+\frac{10064}{24659393841}a^{13}+\frac{1801}{8219797947}a^{12}-\frac{3955}{2739932649}a^{11}+\frac{718}{913310883}a^{10}+\frac{1079}{304436961}a^{9}-\frac{599}{101478987}a^{8}-\frac{160}{33826329}a^{7}+\frac{253}{11275443}a^{6}-\frac{31}{3758481}a^{5}-\frac{74}{1252827}a^{4}+\frac{35}{417609}a^{3}+\frac{13}{139203}a^{2}-\frac{16}{46401}a+\frac{1}{15467}$, $\frac{1}{221934544569}a^{33}-\frac{1}{221934544569}a^{32}-\frac{2}{221934544569}a^{31}+\frac{5}{221934544569}a^{30}+\frac{1}{221934544569}a^{29}-\frac{16}{221934544569}a^{28}+\frac{13}{221934544569}a^{27}+\frac{35}{221934544569}a^{26}-\frac{74}{221934544569}a^{25}-\frac{31}{221934544569}a^{24}+\frac{253}{221934544569}a^{23}-\frac{160}{221934544569}a^{22}-\frac{599}{221934544569}a^{21}+\frac{1079}{221934544569}a^{20}+\frac{718}{221934544569}a^{19}-\frac{2746025}{14348907}a^{18}-\frac{4119037}{14348907}a^{17}-\frac{1991795}{14348907}a^{16}-\frac{1}{14348907}a^{15}+\frac{10064}{73978181523}a^{14}+\frac{1801}{24659393841}a^{13}-\frac{3955}{8219797947}a^{12}+\frac{718}{2739932649}a^{11}+\frac{1079}{913310883}a^{10}-\frac{599}{304436961}a^{9}-\frac{160}{101478987}a^{8}+\frac{253}{33826329}a^{7}-\frac{31}{11275443}a^{6}-\frac{74}{3758481}a^{5}+\frac{35}{1252827}a^{4}+\frac{13}{417609}a^{3}-\frac{16}{139203}a^{2}+\frac{1}{46401}a+\frac{5}{15467}$, $\frac{1}{665803633707}a^{34}-\frac{1}{665803633707}a^{33}-\frac{2}{665803633707}a^{32}+\frac{5}{665803633707}a^{31}+\frac{1}{665803633707}a^{30}-\frac{16}{665803633707}a^{29}+\frac{13}{665803633707}a^{28}+\frac{35}{665803633707}a^{27}-\frac{74}{665803633707}a^{26}-\frac{31}{665803633707}a^{25}+\frac{253}{665803633707}a^{24}-\frac{160}{665803633707}a^{23}-\frac{599}{665803633707}a^{22}+\frac{1079}{665803633707}a^{21}+\frac{718}{665803633707}a^{20}-\frac{3955}{665803633707}a^{19}-\frac{4119037}{43046721}a^{18}+\frac{12357112}{43046721}a^{17}-\frac{1}{43046721}a^{16}+\frac{10064}{221934544569}a^{15}+\frac{1801}{73978181523}a^{14}-\frac{3955}{24659393841}a^{13}+\frac{718}{8219797947}a^{12}+\frac{1079}{2739932649}a^{11}-\frac{599}{913310883}a^{10}-\frac{160}{304436961}a^{9}+\frac{253}{101478987}a^{8}-\frac{31}{33826329}a^{7}-\frac{74}{11275443}a^{6}+\frac{35}{3758481}a^{5}+\frac{13}{1252827}a^{4}-\frac{16}{417609}a^{3}+\frac{1}{139203}a^{2}+\frac{5}{46401}a-\frac{2}{15467}$, $\frac{1}{1997410901121}a^{35}-\frac{1}{1997410901121}a^{34}-\frac{2}{1997410901121}a^{33}+\frac{5}{1997410901121}a^{32}+\frac{1}{1997410901121}a^{31}-\frac{16}{1997410901121}a^{30}+\frac{13}{1997410901121}a^{29}+\frac{35}{1997410901121}a^{28}-\frac{74}{1997410901121}a^{27}-\frac{31}{1997410901121}a^{26}+\frac{253}{1997410901121}a^{25}-\frac{160}{1997410901121}a^{24}-\frac{599}{1997410901121}a^{23}+\frac{1079}{1997410901121}a^{22}+\frac{718}{1997410901121}a^{21}-\frac{3955}{1997410901121}a^{20}+\frac{1801}{1997410901121}a^{19}+\frac{12357112}{129140163}a^{18}-\frac{1}{129140163}a^{17}+\frac{10064}{665803633707}a^{16}+\frac{1801}{221934544569}a^{15}-\frac{3955}{73978181523}a^{14}+\frac{718}{24659393841}a^{13}+\frac{1079}{8219797947}a^{12}-\frac{599}{2739932649}a^{11}-\frac{160}{913310883}a^{10}+\frac{253}{304436961}a^{9}-\frac{31}{101478987}a^{8}-\frac{74}{33826329}a^{7}+\frac{35}{11275443}a^{6}+\frac{13}{3758481}a^{5}-\frac{16}{1252827}a^{4}+\frac{1}{417609}a^{3}+\frac{5}{139203}a^{2}-\frac{2}{46401}a-\frac{1}{15467}$ 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, 1/46401*a^19 + 1/3*a^18 - 1/3*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 - 5403/15467, 1/139203*a^20 - 1/139203*a^19 - 4/9*a^18 + 1/9*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 + 10064/46401*a + 1801/15467, 1/417609*a^21 - 1/417609*a^20 - 2/417609*a^19 - 8/27*a^18 - 7/27*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 + 10064/139203*a^2 + 1801/46401*a - 3955/15467, 1/1252827*a^22 - 1/1252827*a^21 - 2/1252827*a^20 + 5/1252827*a^19 + 20/81*a^18 + 4/81*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 + 10064/417609*a^3 + 1801/139203*a^2 - 3955/46401*a + 718/15467, 1/3758481*a^23 - 1/3758481*a^22 - 2/3758481*a^21 + 5/3758481*a^20 + 1/3758481*a^19 - 77/243*a^18 + 17/243*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 + 10064/1252827*a^4 + 1801/417609*a^3 - 3955/139203*a^2 + 718/46401*a + 1079/15467, 1/11275443*a^24 - 1/11275443*a^23 - 2/11275443*a^22 + 5/11275443*a^21 + 1/11275443*a^20 - 16/11275443*a^19 - 226/729*a^18 - 272/729*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 + 10064/3758481*a^5 + 1801/1252827*a^4 - 3955/417609*a^3 + 718/139203*a^2 + 1079/46401*a - 599/15467, 1/33826329*a^25 - 1/33826329*a^24 - 2/33826329*a^23 + 5/33826329*a^22 + 1/33826329*a^21 - 16/33826329*a^20 + 13/33826329*a^19 + 457/2187*a^18 + 221/2187*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 + 10064/11275443*a^6 + 1801/3758481*a^5 - 3955/1252827*a^4 + 718/417609*a^3 + 1079/139203*a^2 - 599/46401*a - 160/15467, 1/101478987*a^26 - 1/101478987*a^25 - 2/101478987*a^24 + 5/101478987*a^23 + 1/101478987*a^22 - 16/101478987*a^21 + 13/101478987*a^20 + 35/101478987*a^19 + 221/6561*a^18 - 1592/6561*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 + 10064/33826329*a^7 + 1801/11275443*a^6 - 3955/3758481*a^5 + 718/1252827*a^4 + 1079/417609*a^3 - 599/139203*a^2 - 160/46401*a + 253/15467, 1/304436961*a^27 - 1/304436961*a^26 - 2/304436961*a^25 + 5/304436961*a^24 + 1/304436961*a^23 - 16/304436961*a^22 + 13/304436961*a^21 + 35/304436961*a^20 - 74/304436961*a^19 - 1592/19683*a^18 + 929/19683*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 + 10064/101478987*a^8 + 1801/33826329*a^7 - 3955/11275443*a^6 + 718/3758481*a^5 + 1079/1252827*a^4 - 599/417609*a^3 - 160/139203*a^2 + 253/46401*a - 31/15467, 1/913310883*a^28 - 1/913310883*a^27 - 2/913310883*a^26 + 5/913310883*a^25 + 1/913310883*a^24 - 16/913310883*a^23 + 13/913310883*a^22 + 35/913310883*a^21 - 74/913310883*a^20 - 31/913310883*a^19 + 20612/59049*a^18 - 15836/59049*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 + 10064/304436961*a^9 + 1801/101478987*a^8 - 3955/33826329*a^7 + 718/11275443*a^6 + 1079/3758481*a^5 - 599/1252827*a^4 - 160/417609*a^3 + 253/139203*a^2 - 31/46401*a - 74/15467, 1/2739932649*a^29 - 1/2739932649*a^28 - 2/2739932649*a^27 + 5/2739932649*a^26 + 1/2739932649*a^25 - 16/2739932649*a^24 + 13/2739932649*a^23 + 35/2739932649*a^22 - 74/2739932649*a^21 - 31/2739932649*a^20 + 253/2739932649*a^19 + 43213/177147*a^18 + 72098/177147*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 + 10064/913310883*a^10 + 1801/304436961*a^9 - 3955/101478987*a^8 + 718/33826329*a^7 + 1079/11275443*a^6 - 599/3758481*a^5 - 160/1252827*a^4 + 253/417609*a^3 - 31/139203*a^2 - 74/46401*a + 35/15467, 1/8219797947*a^30 - 1/8219797947*a^29 - 2/8219797947*a^28 + 5/8219797947*a^27 + 1/8219797947*a^26 - 16/8219797947*a^25 + 13/8219797947*a^24 + 35/8219797947*a^23 - 74/8219797947*a^22 - 31/8219797947*a^21 + 253/8219797947*a^20 - 160/8219797947*a^19 + 249245/531441*a^18 + 152557/531441*a^17 + 162590/531441*a^16 - 88820/531441*a^15 + 132491/531441*a^14 + 133969/531441*a^13 - 1/531441*a^12 + 10064/2739932649*a^11 + 1801/913310883*a^10 - 3955/304436961*a^9 + 718/101478987*a^8 + 1079/33826329*a^7 - 599/11275443*a^6 - 160/3758481*a^5 + 253/1252827*a^4 - 31/417609*a^3 - 74/139203*a^2 + 35/46401*a + 13/15467, 1/24659393841*a^31 - 1/24659393841*a^30 - 2/24659393841*a^29 + 5/24659393841*a^28 + 1/24659393841*a^27 - 16/24659393841*a^26 + 13/24659393841*a^25 + 35/24659393841*a^24 - 74/24659393841*a^23 - 31/24659393841*a^22 + 253/24659393841*a^21 - 160/24659393841*a^20 - 599/24659393841*a^19 + 152557/1594323*a^18 + 694031/1594323*a^17 + 442621/1594323*a^16 + 663932/1594323*a^15 - 397472/1594323*a^14 - 1/1594323*a^13 + 10064/8219797947*a^12 + 1801/2739932649*a^11 - 3955/913310883*a^10 + 718/304436961*a^9 + 1079/101478987*a^8 - 599/33826329*a^7 - 160/11275443*a^6 + 253/3758481*a^5 - 31/1252827*a^4 - 74/417609*a^3 + 35/139203*a^2 + 13/46401*a - 16/15467, 1/73978181523*a^32 - 1/73978181523*a^31 - 2/73978181523*a^30 + 5/73978181523*a^29 + 1/73978181523*a^28 - 16/73978181523*a^27 + 13/73978181523*a^26 + 35/73978181523*a^25 - 74/73978181523*a^24 - 31/73978181523*a^23 + 253/73978181523*a^22 - 160/73978181523*a^21 - 599/73978181523*a^20 + 1079/73978181523*a^19 + 2288354/4782969*a^18 + 2036944/4782969*a^17 + 663932/4782969*a^16 - 1991795/4782969*a^15 - 1/4782969*a^14 + 10064/24659393841*a^13 + 1801/8219797947*a^12 - 3955/2739932649*a^11 + 718/913310883*a^10 + 1079/304436961*a^9 - 599/101478987*a^8 - 160/33826329*a^7 + 253/11275443*a^6 - 31/3758481*a^5 - 74/1252827*a^4 + 35/417609*a^3 + 13/139203*a^2 - 16/46401*a + 1/15467, 1/221934544569*a^33 - 1/221934544569*a^32 - 2/221934544569*a^31 + 5/221934544569*a^30 + 1/221934544569*a^29 - 16/221934544569*a^28 + 13/221934544569*a^27 + 35/221934544569*a^26 - 74/221934544569*a^25 - 31/221934544569*a^24 + 253/221934544569*a^23 - 160/221934544569*a^22 - 599/221934544569*a^21 + 1079/221934544569*a^20 + 718/221934544569*a^19 - 2746025/14348907*a^18 - 4119037/14348907*a^17 - 1991795/14348907*a^16 - 1/14348907*a^15 + 10064/73978181523*a^14 + 1801/24659393841*a^13 - 3955/8219797947*a^12 + 718/2739932649*a^11 + 1079/913310883*a^10 - 599/304436961*a^9 - 160/101478987*a^8 + 253/33826329*a^7 - 31/11275443*a^6 - 74/3758481*a^5 + 35/1252827*a^4 + 13/417609*a^3 - 16/139203*a^2 + 1/46401*a + 5/15467, 1/665803633707*a^34 - 1/665803633707*a^33 - 2/665803633707*a^32 + 5/665803633707*a^31 + 1/665803633707*a^30 - 16/665803633707*a^29 + 13/665803633707*a^28 + 35/665803633707*a^27 - 74/665803633707*a^26 - 31/665803633707*a^25 + 253/665803633707*a^24 - 160/665803633707*a^23 - 599/665803633707*a^22 + 1079/665803633707*a^21 + 718/665803633707*a^20 - 3955/665803633707*a^19 - 4119037/43046721*a^18 + 12357112/43046721*a^17 - 1/43046721*a^16 + 10064/221934544569*a^15 + 1801/73978181523*a^14 - 3955/24659393841*a^13 + 718/8219797947*a^12 + 1079/2739932649*a^11 - 599/913310883*a^10 - 160/304436961*a^9 + 253/101478987*a^8 - 31/33826329*a^7 - 74/11275443*a^6 + 35/3758481*a^5 + 13/1252827*a^4 - 16/417609*a^3 + 1/139203*a^2 + 5/46401*a - 2/15467, 1/1997410901121*a^35 - 1/1997410901121*a^34 - 2/1997410901121*a^33 + 5/1997410901121*a^32 + 1/1997410901121*a^31 - 16/1997410901121*a^30 + 13/1997410901121*a^29 + 35/1997410901121*a^28 - 74/1997410901121*a^27 - 31/1997410901121*a^26 + 253/1997410901121*a^25 - 160/1997410901121*a^24 - 599/1997410901121*a^23 + 1079/1997410901121*a^22 + 718/1997410901121*a^21 - 3955/1997410901121*a^20 + 1801/1997410901121*a^19 + 12357112/129140163*a^18 - 1/129140163*a^17 + 10064/665803633707*a^16 + 1801/221934544569*a^15 - 3955/73978181523*a^14 + 718/24659393841*a^13 + 1079/8219797947*a^12 - 599/2739932649*a^11 - 160/913310883*a^10 + 253/304436961*a^9 - 31/101478987*a^8 - 74/33826329*a^7 + 35/11275443*a^6 + 13/3758481*a^5 - 16/1252827*a^4 + 1/417609*a^3 + 5/139203*a^2 - 2/46401*a - 1/15467

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:	$17$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( -\frac{10064}{1997410901121} a^{35} + \frac{10064}{1997410901121} a^{34} + \frac{20128}{1997410901121} a^{33} - \frac{50320}{1997410901121} a^{32} - \frac{10064}{1997410901121} a^{31} + \frac{161024}{1997410901121} a^{30} - \frac{130832}{1997410901121} a^{29} - \frac{352240}{1997410901121} a^{28} + \frac{744736}{1997410901121} a^{27} + \frac{311984}{1997410901121} a^{26} - \frac{2546192}{1997410901121} a^{25} + \frac{1610240}{1997410901121} a^{24} + \frac{6028336}{1997410901121} a^{23} - \frac{10859056}{1997410901121} a^{22} - \frac{7225952}{1997410901121} a^{21} + \frac{39803120}{1997410901121} a^{20} - \frac{18125264}{1997410901121} a^{19} + \frac{1801}{129140163} a^{18} + \frac{10064}{129140163} a^{17} - \frac{101284096}{665803633707} a^{16} - \frac{18125264}{221934544569} a^{15} + \frac{39803120}{73978181523} a^{14} - \frac{7225952}{24659393841} a^{13} - \frac{10859056}{8219797947} a^{12} + \frac{6028336}{2739932649} a^{11} + \frac{1610240}{913310883} a^{10} - \frac{2546192}{304436961} a^{9} + \frac{311984}{101478987} a^{8} + \frac{744736}{33826329} a^{7} - \frac{352240}{11275443} a^{6} - \frac{130832}{3758481} a^{5} + \frac{161024}{1252827} a^{4} - \frac{10064}{417609} a^{3} - \frac{50320}{139203} a^{2} + \frac{20128}{46401} a + \frac{10064}{15467} \) -(10064)/(1997410901121)a^(35) + (10064)/(1997410901121)a^(34) + (20128)/(1997410901121)a^(33) - (50320)/(1997410901121)a^(32) - (10064)/(1997410901121)a^(31) + (161024)/(1997410901121)a^(30) - (130832)/(1997410901121)a^(29) - (352240)/(1997410901121)a^(28) + (744736)/(1997410901121)a^(27) + (311984)/(1997410901121)a^(26) - (2546192)/(1997410901121)a^(25) + (1610240)/(1997410901121)a^(24) + (6028336)/(1997410901121)a^(23) - (10859056)/(1997410901121)a^(22) - (7225952)/(1997410901121)a^(21) + (39803120)/(1997410901121)a^(20) - (18125264)/(1997410901121)a^(19) + (1801)/(129140163)a^(18) + (10064)/(129140163)a^(17) - (101284096)/(665803633707)a^(16) - (18125264)/(221934544569)a^(15) + (39803120)/(73978181523)a^(14) - (7225952)/(24659393841)a^(13) - (10859056)/(8219797947)a^(12) + (6028336)/(2739932649)a^(11) + (1610240)/(913310883)a^(10) - (2546192)/(304436961)a^(9) + (311984)/(101478987)a^(8) + (744736)/(33826329)a^(7) - (352240)/(11275443)a^(6) - (130832)/(3758481)a^(5) + (161024)/(1252827)a^(4) - (10064)/(417609)a^(3) - (50320)/(139203)a^(2) + (20128)/(46401)*a + (10064)/(15467) (order $38$)	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^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489)

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^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489, 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^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489);

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^36 - x^35 - 2*x^34 + 5*x^33 + x^32 - 16*x^31 + 13*x^30 + 35*x^29 - 74*x^28 - 31*x^27 + 253*x^26 - 160*x^25 - 599*x^24 + 1079*x^23 + 718*x^22 - 3955*x^21 + 1801*x^20 + 10064*x^19 - 15467*x^18 + 30192*x^17 + 16209*x^16 - 106785*x^15 + 58158*x^14 + 262197*x^13 - 436671*x^12 - 349920*x^11 + 1659933*x^10 - 610173*x^9 - 4369626*x^8 + 6200145*x^7 + 6908733*x^6 - 25509168*x^5 + 4782969*x^4 + 71744535*x^3 - 86093442*x^2 - 129140163*x + 387420489);

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

$C_2\times C_{18}$ (as 36T2):

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 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

$\Q(\sqrt{-19}) $, $\Q(\sqrt{209}) $, $\Q(\sqrt{-11}) $, 3.3.361.1, $\Q(\sqrt{-11}, \sqrt{-19})$, 6.0.2476099.1, 6.6.3295687769.1, 6.0.173457251.1, $\Q(\zeta_{19})^+$, 12.0.10861557870736197361.1, $\Q(\zeta_{19})$, 18.18.12922465537100419689226617716849.1, 18.0.680129765110548404696137774571.1

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	$18^{2}$	$18^{2}$	${\href{/padicField/5.9.0.1}{9} }^{4}$	${\href{/padicField/7.6.0.1}{6} }^{6}$	R	$18^{2}$	$18^{2}$	R	${\href{/padicField/23.9.0.1}{9} }^{4}$	$18^{2}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	${\href{/padicField/47.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$

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	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.2	$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$19$ 19		Deg $36$	$18$	$2$	$34$

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