LMFDB - Number field 44.44.107163231750047944848560960956807209663182497699341984637548299223802110981295940395073536.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089)

gp: K = bnfinit(y^44 - 85*y^42 + 3361*y^40 - 82089*y^38 + 1386817*y^36 - 17197497*y^34 + 162122289*y^32 - 1187271081*y^30 + 6844037217*y^28 - 31279687001*y^26 + 113631570577*y^24 - 327608079881*y^22 + 745527993793*y^20 - 1326378465209*y^18 + 1818601994737*y^16 - 1883097361001*y^14 + 1431629797153*y^12 - 768135683609*y^10 + 274716257617*y^8 - 60026913481*y^6 + 6928067713*y^4 - 316835961*y^2 + 935089, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089)

$ x^{44} - 85 x^{42} + 3361 x^{40} - 82089 x^{38} + 1386817 x^{36} - 17197497 x^{34} + 162122289 x^{32} + \cdots + 935089 $ x^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$44$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[44, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$107\!\cdots\!536$ 107163231750047944848560960956807209663182497699341984637548299223802110981295940395073536 $\medspace = 2^{44}\cdot 7^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$105.54$	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:	$2\cdot 7^{1/2}23^{21/22}\approx 105.5383091233071$
Ramified primes:	$2$, $7$, $23$ 2, 7, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$44$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$644=2^{2}\cdot 7\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(517,·)$, $\chi_{644}(321,·)$, $\chi_{644}(393,·)$, $\chi_{644}(139,·)$, $\chi_{644}(141,·)$, $\chi_{644}(15,·)$, $\chi_{644}(167,·)$, $\chi_{644}(531,·)$, $\chi_{644}(405,·)$, $\chi_{644}(279,·)$, $\chi_{644}(153,·)$, $\chi_{644}(27,·)$, $\chi_{644}(29,·)$, $\chi_{644}(155,·)$, $\chi_{644}(293,·)$, $\chi_{644}(295,·)$, $\chi_{644}(169,·)$, $\chi_{644}(43,·)$, $\chi_{644}(561,·)$, $\chi_{644}(307,·)$, $\chi_{644}(181,·)$, $\chi_{644}(183,·)$, $\chi_{644}(573,·)$, $\chi_{644}(449,·)$, $\chi_{644}(435,·)$, $\chi_{644}(267,·)$, $\chi_{644}(197,·)$, $\chi_{644}(97,·)$, $\chi_{644}(55,·)$, $\chi_{644}(335,·)$, $\chi_{644}(433,·)$, $\chi_{644}(85,·)$, $\chi_{644}(603,·)$, $\chi_{644}(223,·)$, $\chi_{644}(225,·)$, $\chi_{644}(587,·)$, $\chi_{644}(99,·)$, $\chi_{644}(363,·)$, $\chi_{644}(237,·)$, $\chi_{644}(631,·)$, $\chi_{644}(379,·)$, $\chi_{644}(125,·)$, $\chi_{644}(533,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{967}a^{23}-\frac{46}{967}a^{21}-\frac{47}{967}a^{19}+\frac{149}{967}a^{17}-\frac{354}{967}a^{15}-\frac{282}{967}a^{13}+\frac{180}{967}a^{11}+\frac{199}{967}a^{9}+\frac{148}{967}a^{7}-\frac{421}{967}a^{5}-\frac{168}{967}a^{3}+\frac{279}{967}a$, $\frac{1}{5025499}a^{24}+\frac{971789}{5025499}a^{22}+\frac{2469671}{5025499}a^{20}-\frac{1687266}{5025499}a^{18}+\frac{1150376}{5025499}a^{16}-\frac{818364}{5025499}a^{14}+\frac{1124801}{5025499}a^{12}-\frac{994844}{5025499}a^{10}+\frac{622896}{5025499}a^{8}+\frac{219088}{5025499}a^{6}-\frac{844359}{5025499}a^{4}-\frac{1784803}{5025499}a^{2}-\frac{456}{5197}$, $\frac{1}{5025499}a^{25}-\frac{50}{5025499}a^{23}+\frac{1944774}{5025499}a^{21}-\frac{1240324}{5025499}a^{19}+\frac{2085836}{5025499}a^{17}+\frac{1478710}{5025499}a^{15}-\frac{1219046}{5025499}a^{13}-\frac{33399}{5025499}a^{11}-\frac{1804103}{5025499}a^{9}+\frac{2126387}{5025499}a^{7}+\frac{1234441}{5025499}a^{5}+\frac{668181}{5025499}a^{3}-\frac{207087}{5025499}a$, $\frac{1}{5025499}a^{26}+\frac{279234}{5025499}a^{22}+\frac{1631250}{5025499}a^{20}-\frac{1869480}{5025499}a^{18}-\frac{1308478}{5025499}a^{16}-\frac{1933254}{5025499}a^{14}+\frac{926162}{5025499}a^{12}-\frac{1291313}{5025499}a^{10}-\frac{1907306}{5025499}a^{8}+\frac{2137843}{5025499}a^{6}-\frac{1345777}{5025499}a^{4}+\frac{1011745}{5025499}a^{2}-\frac{2012}{5197}$, $\frac{1}{5025499}a^{27}-\frac{1404}{5025499}a^{23}-\frac{535899}{5025499}a^{21}+\frac{1269508}{5025499}a^{19}+\frac{2105951}{5025499}a^{17}+\frac{1928117}{5025499}a^{15}-\frac{341906}{5025499}a^{13}-\frac{1551163}{5025499}a^{11}-\frac{2473779}{5025499}a^{9}+\frac{807411}{5025499}a^{7}+\frac{1216344}{5025499}a^{5}-\frac{2096061}{5025499}a^{3}+\frac{164378}{5025499}a$, $\frac{1}{5025499}a^{28}+\frac{1945628}{5025499}a^{22}+\frac{1093282}{5025499}a^{20}+\frac{194516}{5025499}a^{18}-\frac{1154657}{5025499}a^{16}+\frac{1514309}{5025499}a^{14}-\frac{337245}{5025499}a^{12}-\frac{2146033}{5025499}a^{10}+\frac{916569}{5025499}a^{8}+\frac{2260457}{5025499}a^{6}-\frac{1558333}{5025499}a^{4}+\frac{2024967}{5025499}a^{2}-\frac{993}{5197}$, $\frac{1}{5025499}a^{29}+\frac{1950}{5025499}a^{23}+\frac{43488}{5025499}a^{21}+\frac{1088400}{5025499}a^{19}+\frac{716263}{5025499}a^{17}+\frac{1082958}{5025499}a^{15}+\frac{560}{5025499}a^{13}-\frac{223143}{5025499}a^{11}+\frac{1088070}{5025499}a^{9}+\frac{1049556}{5025499}a^{7}-\frac{2426232}{5025499}a^{5}+\frac{1905436}{5025499}a^{3}-\frac{492501}{5025499}a$, $\frac{1}{5025499}a^{30}-\frac{331939}{5025499}a^{22}-\frac{342008}{5025499}a^{20}-\frac{816882}{5025499}a^{18}-\frac{777688}{5025499}a^{16}-\frac{2298322}{5025499}a^{14}-\frac{2467529}{5025499}a^{12}+\frac{1191256}{5025499}a^{10}-\frac{2452385}{5025499}a^{8}-\frac{2480417}{5025499}a^{6}+\frac{41814}{5025499}a^{4}+\frac{2228041}{5025499}a^{2}+\frac{513}{5197}$, $\frac{1}{5025499}a^{31}+\frac{669}{5025499}a^{23}-\frac{565479}{5025499}a^{21}-\frac{1372961}{5025499}a^{19}-\frac{1474086}{5025499}a^{17}+\frac{570422}{5025499}a^{15}-\frac{778504}{5025499}a^{13}+\frac{754708}{5025499}a^{11}-\frac{1594880}{5025499}a^{9}+\frac{1516076}{5025499}a^{7}+\frac{727818}{5025499}a^{5}+\frac{1630386}{5025499}a^{3}-\frac{2190778}{5025499}a$, $\frac{1}{5025499}a^{32}-\frac{2402949}{5025499}a^{22}-\frac{193689}{5025499}a^{20}+\frac{1595092}{5025499}a^{18}-\frac{129775}{5025499}a^{16}-\frac{1072379}{5025499}a^{14}+\frac{2087689}{5025499}a^{12}+\frac{589888}{5025499}a^{10}+\frac{1915069}{5025499}a^{8}-\frac{102583}{5025499}a^{6}-\frac{1374830}{5025499}a^{4}+\frac{799166}{5025499}a^{2}-\frac{1559}{5197}$, $\frac{1}{5025499}a^{33}-\frac{1935}{5025499}a^{23}-\frac{79355}{5025499}a^{21}-\frac{691588}{5025499}a^{19}+\frac{810882}{5025499}a^{17}-\frac{1722004}{5025499}a^{15}-\frac{1581393}{5025499}a^{13}+\frac{579494}{5025499}a^{11}+\frac{2294450}{5025499}a^{9}-\frac{1562940}{5025499}a^{7}-\frac{2076425}{5025499}a^{5}-\frac{531266}{5025499}a^{3}-\frac{16014}{5025499}a$, $\frac{1}{5025499}a^{34}+\frac{795734}{5025499}a^{22}-\frac{1127752}{5025499}a^{20}-\frac{2499977}{5025499}a^{18}-\frac{2040501}{5025499}a^{16}-\frac{2083548}{5025499}a^{14}+\frac{1028362}{5025499}a^{12}+\frac{2037427}{5025499}a^{10}-\frac{2378940}{5025499}a^{8}-\frac{283061}{5025499}a^{6}-\frac{1078756}{5025499}a^{4}-\frac{1092006}{5025499}a^{2}+\frac{1130}{5197}$, $\frac{1}{5025499}a^{35}+\frac{593}{5025499}a^{23}+\frac{270241}{5025499}a^{21}-\frac{306843}{5025499}a^{19}+\frac{95466}{5025499}a^{17}-\frac{2031578}{5025499}a^{15}-\frac{889331}{5025499}a^{13}-\frac{373981}{5025499}a^{11}+\frac{203969}{5025499}a^{9}-\frac{2377452}{5025499}a^{7}+\frac{1992671}{5025499}a^{5}+\frac{1828708}{5025499}a^{3}+\frac{370327}{5025499}a$, $\frac{1}{5025499}a^{36}+\frac{1931749}{5025499}a^{22}-\frac{2401537}{5025499}a^{20}+\frac{569903}{5025499}a^{18}-\frac{736682}{5025499}a^{16}+\frac{1952617}{5025499}a^{14}+\frac{1010393}{5025499}a^{12}+\frac{2163078}{5025499}a^{10}+\frac{132146}{5025499}a^{8}-\frac{2289038}{5025499}a^{6}-\frac{16305}{5025499}a^{4}-\frac{1621783}{5025499}a^{2}+\frac{164}{5197}$, $\frac{1}{5025499}a^{37}-\frac{1535}{5025499}a^{23}+\frac{1096044}{5025499}a^{21}+\frac{975269}{5025499}a^{19}-\frac{2342555}{5025499}a^{17}-\frac{2158210}{5025499}a^{15}-\frac{1582910}{5025499}a^{13}+\frac{931389}{5025499}a^{11}+\frac{2372053}{5025499}a^{9}-\frac{1961627}{5025499}a^{7}-\frac{234579}{5025499}a^{5}+\frac{1537993}{5025499}a^{3}-\frac{1499255}{5025499}a$, $\frac{1}{5025499}a^{38}+\frac{218956}{5025499}a^{22}-\frac{2331491}{5025499}a^{20}+\frac{861619}{5025499}a^{18}-\frac{281199}{5025499}a^{16}-\frac{1396900}{5025499}a^{14}-\frac{1270732}{5025499}a^{12}-\frac{1987290}{5025499}a^{10}-\frac{661077}{5025499}a^{8}-\frac{642932}{5025499}a^{6}+\frac{2025670}{5025499}a^{4}-\frac{2274905}{5025499}a^{2}+\frac{1635}{5197}$, $\frac{1}{5025499}a^{39}+\frac{682}{5025499}a^{23}-\frac{2341885}{5025499}a^{21}+\frac{1069499}{5025499}a^{19}+\frac{2374468}{5025499}a^{17}+\frac{489611}{5025499}a^{15}-\frac{23452}{5025499}a^{13}-\frac{1072618}{5025499}a^{11}+\frac{1131888}{5025499}a^{9}+\frac{2231009}{5025499}a^{7}-\frac{1565457}{5025499}a^{5}-\frac{783366}{5025499}a^{3}+\frac{988587}{5025499}a$, $\frac{1}{5025499}a^{40}-\frac{1736115}{5025499}a^{22}+\frac{296042}{5025499}a^{20}+\frac{2250609}{5025499}a^{18}-\frac{88977}{5025499}a^{16}+\frac{270407}{5025499}a^{14}+\frac{714447}{5025499}a^{12}+\frac{1173131}{5025499}a^{10}-\frac{442147}{5025499}a^{8}-\frac{218503}{5025499}a^{6}+\frac{2162586}{5025499}a^{4}+\frac{2053475}{5025499}a^{2}-\frac{828}{5197}$, $\frac{1}{5025499}a^{41}-\frac{317}{5025499}a^{23}+\frac{857318}{5025499}a^{21}+\frac{1076087}{5025499}a^{19}+\frac{2244476}{5025499}a^{17}-\frac{1091207}{5025499}a^{15}-\frac{1307186}{5025499}a^{13}+\frac{2035833}{5025499}a^{11}-\frac{1777776}{5025499}a^{9}+\frac{379152}{5025499}a^{7}+\frac{88983}{5025499}a^{5}+\frac{1918353}{5025499}a^{3}+\frac{1039062}{5025499}a$, $\frac{1}{5025499}a^{42}+\frac{2358992}{5025499}a^{22}-\frac{16050}{5025499}a^{20}+\frac{84048}{5025499}a^{18}+\frac{1742057}{5025499}a^{16}+\frac{597374}{5025499}a^{14}+\frac{1787321}{5025499}a^{12}-\frac{536887}{5025499}a^{10}+\frac{1842723}{5025499}a^{8}-\frac{817107}{5025499}a^{6}+\frac{607997}{5025499}a^{4}-\frac{1887601}{5025499}a^{2}+\frac{964}{5197}$, $\frac{1}{5025499}a^{43}-\frac{446}{5025499}a^{23}-\frac{2042880}{5025499}a^{21}+\frac{416656}{5025499}a^{19}+\frac{1970725}{5025499}a^{17}+\frac{1605592}{5025499}a^{15}-\frac{1242530}{5025499}a^{13}+\frac{1931688}{5025499}a^{11}-\frac{314032}{5025499}a^{9}+\frac{1770999}{5025499}a^{7}-\frac{1117407}{5025499}a^{5}+\frac{2509061}{5025499}a^{3}+\frac{989355}{5025499}a$ 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, 1/967*a^23 - 46/967*a^21 - 47/967*a^19 + 149/967*a^17 - 354/967*a^15 - 282/967*a^13 + 180/967*a^11 + 199/967*a^9 + 148/967*a^7 - 421/967*a^5 - 168/967*a^3 + 279/967*a, 1/5025499*a^24 + 971789/5025499*a^22 + 2469671/5025499*a^20 - 1687266/5025499*a^18 + 1150376/5025499*a^16 - 818364/5025499*a^14 + 1124801/5025499*a^12 - 994844/5025499*a^10 + 622896/5025499*a^8 + 219088/5025499*a^6 - 844359/5025499*a^4 - 1784803/5025499*a^2 - 456/5197, 1/5025499*a^25 - 50/5025499*a^23 + 1944774/5025499*a^21 - 1240324/5025499*a^19 + 2085836/5025499*a^17 + 1478710/5025499*a^15 - 1219046/5025499*a^13 - 33399/5025499*a^11 - 1804103/5025499*a^9 + 2126387/5025499*a^7 + 1234441/5025499*a^5 + 668181/5025499*a^3 - 207087/5025499*a, 1/5025499*a^26 + 279234/5025499*a^22 + 1631250/5025499*a^20 - 1869480/5025499*a^18 - 1308478/5025499*a^16 - 1933254/5025499*a^14 + 926162/5025499*a^12 - 1291313/5025499*a^10 - 1907306/5025499*a^8 + 2137843/5025499*a^6 - 1345777/5025499*a^4 + 1011745/5025499*a^2 - 2012/5197, 1/5025499*a^27 - 1404/5025499*a^23 - 535899/5025499*a^21 + 1269508/5025499*a^19 + 2105951/5025499*a^17 + 1928117/5025499*a^15 - 341906/5025499*a^13 - 1551163/5025499*a^11 - 2473779/5025499*a^9 + 807411/5025499*a^7 + 1216344/5025499*a^5 - 2096061/5025499*a^3 + 164378/5025499*a, 1/5025499*a^28 + 1945628/5025499*a^22 + 1093282/5025499*a^20 + 194516/5025499*a^18 - 1154657/5025499*a^16 + 1514309/5025499*a^14 - 337245/5025499*a^12 - 2146033/5025499*a^10 + 916569/5025499*a^8 + 2260457/5025499*a^6 - 1558333/5025499*a^4 + 2024967/5025499*a^2 - 993/5197, 1/5025499*a^29 + 1950/5025499*a^23 + 43488/5025499*a^21 + 1088400/5025499*a^19 + 716263/5025499*a^17 + 1082958/5025499*a^15 + 560/5025499*a^13 - 223143/5025499*a^11 + 1088070/5025499*a^9 + 1049556/5025499*a^7 - 2426232/5025499*a^5 + 1905436/5025499*a^3 - 492501/5025499*a, 1/5025499*a^30 - 331939/5025499*a^22 - 342008/5025499*a^20 - 816882/5025499*a^18 - 777688/5025499*a^16 - 2298322/5025499*a^14 - 2467529/5025499*a^12 + 1191256/5025499*a^10 - 2452385/5025499*a^8 - 2480417/5025499*a^6 + 41814/5025499*a^4 + 2228041/5025499*a^2 + 513/5197, 1/5025499*a^31 + 669/5025499*a^23 - 565479/5025499*a^21 - 1372961/5025499*a^19 - 1474086/5025499*a^17 + 570422/5025499*a^15 - 778504/5025499*a^13 + 754708/5025499*a^11 - 1594880/5025499*a^9 + 1516076/5025499*a^7 + 727818/5025499*a^5 + 1630386/5025499*a^3 - 2190778/5025499*a, 1/5025499*a^32 - 2402949/5025499*a^22 - 193689/5025499*a^20 + 1595092/5025499*a^18 - 129775/5025499*a^16 - 1072379/5025499*a^14 + 2087689/5025499*a^12 + 589888/5025499*a^10 + 1915069/5025499*a^8 - 102583/5025499*a^6 - 1374830/5025499*a^4 + 799166/5025499*a^2 - 1559/5197, 1/5025499*a^33 - 1935/5025499*a^23 - 79355/5025499*a^21 - 691588/5025499*a^19 + 810882/5025499*a^17 - 1722004/5025499*a^15 - 1581393/5025499*a^13 + 579494/5025499*a^11 + 2294450/5025499*a^9 - 1562940/5025499*a^7 - 2076425/5025499*a^5 - 531266/5025499*a^3 - 16014/5025499*a, 1/5025499*a^34 + 795734/5025499*a^22 - 1127752/5025499*a^20 - 2499977/5025499*a^18 - 2040501/5025499*a^16 - 2083548/5025499*a^14 + 1028362/5025499*a^12 + 2037427/5025499*a^10 - 2378940/5025499*a^8 - 283061/5025499*a^6 - 1078756/5025499*a^4 - 1092006/5025499*a^2 + 1130/5197, 1/5025499*a^35 + 593/5025499*a^23 + 270241/5025499*a^21 - 306843/5025499*a^19 + 95466/5025499*a^17 - 2031578/5025499*a^15 - 889331/5025499*a^13 - 373981/5025499*a^11 + 203969/5025499*a^9 - 2377452/5025499*a^7 + 1992671/5025499*a^5 + 1828708/5025499*a^3 + 370327/5025499*a, 1/5025499*a^36 + 1931749/5025499*a^22 - 2401537/5025499*a^20 + 569903/5025499*a^18 - 736682/5025499*a^16 + 1952617/5025499*a^14 + 1010393/5025499*a^12 + 2163078/5025499*a^10 + 132146/5025499*a^8 - 2289038/5025499*a^6 - 16305/5025499*a^4 - 1621783/5025499*a^2 + 164/5197, 1/5025499*a^37 - 1535/5025499*a^23 + 1096044/5025499*a^21 + 975269/5025499*a^19 - 2342555/5025499*a^17 - 2158210/5025499*a^15 - 1582910/5025499*a^13 + 931389/5025499*a^11 + 2372053/5025499*a^9 - 1961627/5025499*a^7 - 234579/5025499*a^5 + 1537993/5025499*a^3 - 1499255/5025499*a, 1/5025499*a^38 + 218956/5025499*a^22 - 2331491/5025499*a^20 + 861619/5025499*a^18 - 281199/5025499*a^16 - 1396900/5025499*a^14 - 1270732/5025499*a^12 - 1987290/5025499*a^10 - 661077/5025499*a^8 - 642932/5025499*a^6 + 2025670/5025499*a^4 - 2274905/5025499*a^2 + 1635/5197, 1/5025499*a^39 + 682/5025499*a^23 - 2341885/5025499*a^21 + 1069499/5025499*a^19 + 2374468/5025499*a^17 + 489611/5025499*a^15 - 23452/5025499*a^13 - 1072618/5025499*a^11 + 1131888/5025499*a^9 + 2231009/5025499*a^7 - 1565457/5025499*a^5 - 783366/5025499*a^3 + 988587/5025499*a, 1/5025499*a^40 - 1736115/5025499*a^22 + 296042/5025499*a^20 + 2250609/5025499*a^18 - 88977/5025499*a^16 + 270407/5025499*a^14 + 714447/5025499*a^12 + 1173131/5025499*a^10 - 442147/5025499*a^8 - 218503/5025499*a^6 + 2162586/5025499*a^4 + 2053475/5025499*a^2 - 828/5197, 1/5025499*a^41 - 317/5025499*a^23 + 857318/5025499*a^21 + 1076087/5025499*a^19 + 2244476/5025499*a^17 - 1091207/5025499*a^15 - 1307186/5025499*a^13 + 2035833/5025499*a^11 - 1777776/5025499*a^9 + 379152/5025499*a^7 + 88983/5025499*a^5 + 1918353/5025499*a^3 + 1039062/5025499*a, 1/5025499*a^42 + 2358992/5025499*a^22 - 16050/5025499*a^20 + 84048/5025499*a^18 + 1742057/5025499*a^16 + 597374/5025499*a^14 + 1787321/5025499*a^12 - 536887/5025499*a^10 + 1842723/5025499*a^8 - 817107/5025499*a^6 + 607997/5025499*a^4 - 1887601/5025499*a^2 + 964/5197, 1/5025499*a^43 - 446/5025499*a^23 - 2042880/5025499*a^21 + 416656/5025499*a^19 + 1970725/5025499*a^17 + 1605592/5025499*a^15 - 1242530/5025499*a^13 + 1931688/5025499*a^11 - 314032/5025499*a^9 + 1770999/5025499*a^7 - 1117407/5025499*a^5 + 2509061/5025499*a^3 + 989355/5025499*a

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:	$43$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089)

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^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089, 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^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089);

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^44 - 85*x^42 + 3361*x^40 - 82089*x^38 + 1386817*x^36 - 17197497*x^34 + 162122289*x^32 - 1187271081*x^30 + 6844037217*x^28 - 31279687001*x^26 + 113631570577*x^24 - 327608079881*x^22 + 745527993793*x^20 - 1326378465209*x^18 + 1818601994737*x^16 - 1883097361001*x^14 + 1431629797153*x^12 - 768135683609*x^10 + 274716257617*x^8 - 60026913481*x^6 + 6928067713*x^4 - 316835961*x^2 + 935089);

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

$C_2\times C_{22}$ (as 44T2):

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 44

The 44 conjugacy class representatives for $C_2\times C_{22}$

Character table for $C_2\times C_{22}$

Intermediate fields

$\Q(\sqrt{161}) $, $\Q(\sqrt{23}) $, $\Q(\sqrt{7}) $, $\Q(\sqrt{7}, \sqrt{23})$, $\Q(\zeta_{23})^+$, 22.22.78048218870425324004237696277333187889.1, $\Q(\zeta_{92})^+$, 22.22.14232954634830452964011747236817552143286272.1

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	R	$22^{2}$	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/19.11.0.1}{11} }^{4}$	R	${\href{/padicField/29.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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
$2$ 2		Deg $22$	$2$	$11$	$22$
$2$ 2		Deg $22$	$2$	$11$	$22$
$7$ 7	7.22.11.1	$x^{22} + 282475249 x^{2} - 7909306972$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
$7$ 7	7.22.11.1	$x^{22} + 282475249 x^{2} - 7909306972$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
$23$ 23		Deg $44$	$22$	$2$	$42$

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