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Properties

Label 44.0.126...409.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.268\times 10^{80}$
Root discriminant \(66.15\)
Ramified primes $11,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609)
 
gp: K = bnfinit(y^44 - y^43 - 2*y^42 + 5*y^41 + y^40 - 16*y^39 + 13*y^38 + 35*y^37 - 74*y^36 - 31*y^35 + 253*y^34 - 160*y^33 - 599*y^32 + 1079*y^31 + 718*y^30 - 3955*y^29 + 1801*y^28 + 10064*y^27 - 15467*y^26 - 14725*y^25 + 61126*y^24 - 16951*y^23 - 166427*y^22 - 50853*y^21 + 550134*y^20 - 397575*y^19 - 1252827*y^18 + 2445552*y^17 + 1312929*y^16 - 8649585*y^15 + 4710798*y^14 + 21237957*y^13 - 35370351*y^12 - 28343520*y^11 + 134454573*y^10 - 49424013*y^9 - 353939706*y^8 + 502211745*y^7 + 559607373*y^6 - 2066242608*y^5 + 387420489*y^4 + 5811307335*y^3 - 6973568802*y^2 - 10460353203*y + 31381059609, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609)
 

\( x^{44} - x^{43} - 2 x^{42} + 5 x^{41} + x^{40} - 16 x^{39} + 13 x^{38} + 35 x^{37} - 74 x^{36} + \cdots + 31381059609 \) Copy content Toggle raw display

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:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(126\!\cdots\!409\) \(\medspace = 11^{22}\cdot 23^{42}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(66.15\)
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:  $11^{1/2}23^{21/22}\approx 66.14963600447149$
Ramified primes:   \(11\), \(23\) Copy content Toggle raw display
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:  \(253=11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{253}(1,·)$, $\chi_{253}(131,·)$, $\chi_{253}(133,·)$, $\chi_{253}(10,·)$, $\chi_{253}(12,·)$, $\chi_{253}(142,·)$, $\chi_{253}(144,·)$, $\chi_{253}(21,·)$, $\chi_{253}(153,·)$, $\chi_{253}(155,·)$, $\chi_{253}(32,·)$, $\chi_{253}(34,·)$, $\chi_{253}(164,·)$, $\chi_{253}(166,·)$, $\chi_{253}(43,·)$, $\chi_{253}(45,·)$, $\chi_{253}(175,·)$, $\chi_{253}(177,·)$, $\chi_{253}(54,·)$, $\chi_{253}(56,·)$, $\chi_{253}(186,·)$, $\chi_{253}(188,·)$, $\chi_{253}(65,·)$, $\chi_{253}(67,·)$, $\chi_{253}(197,·)$, $\chi_{253}(199,·)$, $\chi_{253}(76,·)$, $\chi_{253}(78,·)$, $\chi_{253}(208,·)$, $\chi_{253}(210,·)$, $\chi_{253}(87,·)$, $\chi_{253}(89,·)$, $\chi_{253}(219,·)$, $\chi_{253}(221,·)$, $\chi_{253}(98,·)$, $\chi_{253}(100,·)$, $\chi_{253}(232,·)$, $\chi_{253}(109,·)$, $\chi_{253}(111,·)$, $\chi_{253}(241,·)$, $\chi_{253}(243,·)$, $\chi_{253}(120,·)$, $\chi_{253}(122,·)$, $\chi_{253}(252,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{499281}a^{23}+\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{16951}{166427}$, $\frac{1}{1497843}a^{24}-\frac{1}{1497843}a^{23}+\frac{2}{9}a^{22}+\frac{4}{9}a^{21}-\frac{1}{9}a^{20}-\frac{2}{9}a^{19}-\frac{4}{9}a^{18}+\frac{1}{9}a^{17}+\frac{2}{9}a^{16}+\frac{4}{9}a^{15}-\frac{1}{9}a^{14}-\frac{2}{9}a^{13}-\frac{4}{9}a^{12}+\frac{1}{9}a^{11}+\frac{2}{9}a^{10}+\frac{4}{9}a^{9}-\frac{1}{9}a^{8}-\frac{2}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{2}{9}a^{4}+\frac{4}{9}a^{3}-\frac{1}{9}a^{2}-\frac{16951}{499281}a+\frac{61126}{166427}$, $\frac{1}{4493529}a^{25}-\frac{1}{4493529}a^{24}-\frac{2}{4493529}a^{23}-\frac{5}{27}a^{22}-\frac{1}{27}a^{21}-\frac{11}{27}a^{20}-\frac{13}{27}a^{19}-\frac{8}{27}a^{18}-\frac{7}{27}a^{17}+\frac{4}{27}a^{16}-\frac{10}{27}a^{15}-\frac{2}{27}a^{14}+\frac{5}{27}a^{13}+\frac{1}{27}a^{12}+\frac{11}{27}a^{11}+\frac{13}{27}a^{10}+\frac{8}{27}a^{9}+\frac{7}{27}a^{8}-\frac{4}{27}a^{7}+\frac{10}{27}a^{6}+\frac{2}{27}a^{5}-\frac{5}{27}a^{4}-\frac{1}{27}a^{3}-\frac{16951}{1497843}a^{2}+\frac{61126}{499281}a-\frac{14725}{166427}$, $\frac{1}{13480587}a^{26}-\frac{1}{13480587}a^{25}-\frac{2}{13480587}a^{24}+\frac{5}{13480587}a^{23}+\frac{26}{81}a^{22}-\frac{11}{81}a^{21}+\frac{14}{81}a^{20}+\frac{19}{81}a^{19}+\frac{20}{81}a^{18}+\frac{4}{81}a^{17}+\frac{17}{81}a^{16}-\frac{29}{81}a^{15}-\frac{22}{81}a^{14}+\frac{28}{81}a^{13}+\frac{38}{81}a^{12}+\frac{40}{81}a^{11}+\frac{8}{81}a^{10}+\frac{34}{81}a^{9}+\frac{23}{81}a^{8}+\frac{37}{81}a^{7}-\frac{25}{81}a^{6}-\frac{5}{81}a^{5}-\frac{1}{81}a^{4}-\frac{16951}{4493529}a^{3}+\frac{61126}{1497843}a^{2}-\frac{14725}{499281}a-\frac{15467}{166427}$, $\frac{1}{40441761}a^{27}-\frac{1}{40441761}a^{26}-\frac{2}{40441761}a^{25}+\frac{5}{40441761}a^{24}+\frac{1}{40441761}a^{23}+\frac{70}{243}a^{22}+\frac{95}{243}a^{21}-\frac{62}{243}a^{20}+\frac{20}{243}a^{19}-\frac{77}{243}a^{18}+\frac{17}{243}a^{17}-\frac{29}{243}a^{16}-\frac{22}{243}a^{15}+\frac{109}{243}a^{14}-\frac{43}{243}a^{13}-\frac{41}{243}a^{12}-\frac{73}{243}a^{11}-\frac{47}{243}a^{10}+\frac{23}{243}a^{9}+\frac{118}{243}a^{8}+\frac{56}{243}a^{7}+\frac{76}{243}a^{6}-\frac{1}{243}a^{5}-\frac{16951}{13480587}a^{4}+\frac{61126}{4493529}a^{3}-\frac{14725}{1497843}a^{2}-\frac{15467}{499281}a+\frac{10064}{166427}$, $\frac{1}{121325283}a^{28}-\frac{1}{121325283}a^{27}-\frac{2}{121325283}a^{26}+\frac{5}{121325283}a^{25}+\frac{1}{121325283}a^{24}-\frac{16}{121325283}a^{23}+\frac{95}{729}a^{22}-\frac{305}{729}a^{21}+\frac{20}{729}a^{20}+\frac{166}{729}a^{19}-\frac{226}{729}a^{18}-\frac{272}{729}a^{17}+\frac{221}{729}a^{16}-\frac{134}{729}a^{15}+\frac{200}{729}a^{14}+\frac{202}{729}a^{13}-\frac{73}{729}a^{12}+\frac{196}{729}a^{11}+\frac{23}{729}a^{10}+\frac{118}{729}a^{9}-\frac{187}{729}a^{8}-\frac{167}{729}a^{7}-\frac{1}{729}a^{6}-\frac{16951}{40441761}a^{5}+\frac{61126}{13480587}a^{4}-\frac{14725}{4493529}a^{3}-\frac{15467}{1497843}a^{2}+\frac{10064}{499281}a+\frac{1801}{166427}$, $\frac{1}{363975849}a^{29}-\frac{1}{363975849}a^{28}-\frac{2}{363975849}a^{27}+\frac{5}{363975849}a^{26}+\frac{1}{363975849}a^{25}-\frac{16}{363975849}a^{24}+\frac{13}{363975849}a^{23}-\frac{305}{2187}a^{22}+\frac{20}{2187}a^{21}+\frac{895}{2187}a^{20}-\frac{955}{2187}a^{19}+\frac{457}{2187}a^{18}+\frac{221}{2187}a^{17}+\frac{595}{2187}a^{16}+\frac{929}{2187}a^{15}-\frac{527}{2187}a^{14}-\frac{73}{2187}a^{13}-\frac{533}{2187}a^{12}+\frac{752}{2187}a^{11}+\frac{847}{2187}a^{10}-\frac{916}{2187}a^{9}+\frac{562}{2187}a^{8}-\frac{1}{2187}a^{7}-\frac{16951}{121325283}a^{6}+\frac{61126}{40441761}a^{5}-\frac{14725}{13480587}a^{4}-\frac{15467}{4493529}a^{3}+\frac{10064}{1497843}a^{2}+\frac{1801}{499281}a-\frac{3955}{166427}$, $\frac{1}{1091927547}a^{30}-\frac{1}{1091927547}a^{29}-\frac{2}{1091927547}a^{28}+\frac{5}{1091927547}a^{27}+\frac{1}{1091927547}a^{26}-\frac{16}{1091927547}a^{25}+\frac{13}{1091927547}a^{24}+\frac{35}{1091927547}a^{23}+\frac{2207}{6561}a^{22}-\frac{1292}{6561}a^{21}+\frac{1232}{6561}a^{20}+\frac{2644}{6561}a^{19}+\frac{221}{6561}a^{18}-\frac{1592}{6561}a^{17}+\frac{929}{6561}a^{16}-\frac{2714}{6561}a^{15}-\frac{73}{6561}a^{14}+\frac{1654}{6561}a^{13}-\frac{1435}{6561}a^{12}+\frac{3034}{6561}a^{11}+\frac{1271}{6561}a^{10}+\frac{2749}{6561}a^{9}-\frac{1}{6561}a^{8}-\frac{16951}{363975849}a^{7}+\frac{61126}{121325283}a^{6}-\frac{14725}{40441761}a^{5}-\frac{15467}{13480587}a^{4}+\frac{10064}{4493529}a^{3}+\frac{1801}{1497843}a^{2}-\frac{3955}{499281}a+\frac{718}{166427}$, $\frac{1}{3275782641}a^{31}-\frac{1}{3275782641}a^{30}-\frac{2}{3275782641}a^{29}+\frac{5}{3275782641}a^{28}+\frac{1}{3275782641}a^{27}-\frac{16}{3275782641}a^{26}+\frac{13}{3275782641}a^{25}+\frac{35}{3275782641}a^{24}-\frac{74}{3275782641}a^{23}-\frac{7853}{19683}a^{22}+\frac{1232}{19683}a^{21}+\frac{2644}{19683}a^{20}-\frac{6340}{19683}a^{19}-\frac{1592}{19683}a^{18}+\frac{929}{19683}a^{17}+\frac{3847}{19683}a^{16}-\frac{6634}{19683}a^{15}-\frac{4907}{19683}a^{14}+\frac{5126}{19683}a^{13}+\frac{9595}{19683}a^{12}-\frac{5290}{19683}a^{11}-\frac{3812}{19683}a^{10}-\frac{1}{19683}a^{9}-\frac{16951}{1091927547}a^{8}+\frac{61126}{363975849}a^{7}-\frac{14725}{121325283}a^{6}-\frac{15467}{40441761}a^{5}+\frac{10064}{13480587}a^{4}+\frac{1801}{4493529}a^{3}-\frac{3955}{1497843}a^{2}+\frac{718}{499281}a+\frac{1079}{166427}$, $\frac{1}{9827347923}a^{32}-\frac{1}{9827347923}a^{31}-\frac{2}{9827347923}a^{30}+\frac{5}{9827347923}a^{29}+\frac{1}{9827347923}a^{28}-\frac{16}{9827347923}a^{27}+\frac{13}{9827347923}a^{26}+\frac{35}{9827347923}a^{25}-\frac{74}{9827347923}a^{24}-\frac{31}{9827347923}a^{23}+\frac{20915}{59049}a^{22}+\frac{2644}{59049}a^{21}-\frac{6340}{59049}a^{20}-\frac{1592}{59049}a^{19}+\frac{20612}{59049}a^{18}-\frac{15836}{59049}a^{17}+\frac{13049}{59049}a^{16}-\frac{24590}{59049}a^{15}-\frac{14557}{59049}a^{14}+\frac{29278}{59049}a^{13}+\frac{14393}{59049}a^{12}+\frac{15871}{59049}a^{11}-\frac{1}{59049}a^{10}-\frac{16951}{3275782641}a^{9}+\frac{61126}{1091927547}a^{8}-\frac{14725}{363975849}a^{7}-\frac{15467}{121325283}a^{6}+\frac{10064}{40441761}a^{5}+\frac{1801}{13480587}a^{4}-\frac{3955}{4493529}a^{3}+\frac{718}{1497843}a^{2}+\frac{1079}{499281}a-\frac{599}{166427}$, $\frac{1}{29482043769}a^{33}-\frac{1}{29482043769}a^{32}-\frac{2}{29482043769}a^{31}+\frac{5}{29482043769}a^{30}+\frac{1}{29482043769}a^{29}-\frac{16}{29482043769}a^{28}+\frac{13}{29482043769}a^{27}+\frac{35}{29482043769}a^{26}-\frac{74}{29482043769}a^{25}-\frac{31}{29482043769}a^{24}+\frac{253}{29482043769}a^{23}+\frac{2644}{177147}a^{22}-\frac{65389}{177147}a^{21}+\frac{57457}{177147}a^{20}-\frac{38437}{177147}a^{19}+\frac{43213}{177147}a^{18}+\frac{72098}{177147}a^{17}-\frac{24590}{177147}a^{16}-\frac{14557}{177147}a^{15}+\frac{88327}{177147}a^{14}-\frac{44656}{177147}a^{13}-\frac{43178}{177147}a^{12}-\frac{1}{177147}a^{11}-\frac{16951}{9827347923}a^{10}+\frac{61126}{3275782641}a^{9}-\frac{14725}{1091927547}a^{8}-\frac{15467}{363975849}a^{7}+\frac{10064}{121325283}a^{6}+\frac{1801}{40441761}a^{5}-\frac{3955}{13480587}a^{4}+\frac{718}{4493529}a^{3}+\frac{1079}{1497843}a^{2}-\frac{599}{499281}a-\frac{160}{166427}$, $\frac{1}{88446131307}a^{34}-\frac{1}{88446131307}a^{33}-\frac{2}{88446131307}a^{32}+\frac{5}{88446131307}a^{31}+\frac{1}{88446131307}a^{30}-\frac{16}{88446131307}a^{29}+\frac{13}{88446131307}a^{28}+\frac{35}{88446131307}a^{27}-\frac{74}{88446131307}a^{26}-\frac{31}{88446131307}a^{25}+\frac{253}{88446131307}a^{24}-\frac{160}{88446131307}a^{23}-\frac{242536}{531441}a^{22}+\frac{234604}{531441}a^{21}-\frac{38437}{531441}a^{20}-\frac{133934}{531441}a^{19}+\frac{249245}{531441}a^{18}+\frac{152557}{531441}a^{17}+\frac{162590}{531441}a^{16}-\frac{88820}{531441}a^{15}+\frac{132491}{531441}a^{14}+\frac{133969}{531441}a^{13}-\frac{1}{531441}a^{12}-\frac{16951}{29482043769}a^{11}+\frac{61126}{9827347923}a^{10}-\frac{14725}{3275782641}a^{9}-\frac{15467}{1091927547}a^{8}+\frac{10064}{363975849}a^{7}+\frac{1801}{121325283}a^{6}-\frac{3955}{40441761}a^{5}+\frac{718}{13480587}a^{4}+\frac{1079}{4493529}a^{3}-\frac{599}{1497843}a^{2}-\frac{160}{499281}a+\frac{253}{166427}$, $\frac{1}{265338393921}a^{35}-\frac{1}{265338393921}a^{34}-\frac{2}{265338393921}a^{33}+\frac{5}{265338393921}a^{32}+\frac{1}{265338393921}a^{31}-\frac{16}{265338393921}a^{30}+\frac{13}{265338393921}a^{29}+\frac{35}{265338393921}a^{28}-\frac{74}{265338393921}a^{27}-\frac{31}{265338393921}a^{26}+\frac{253}{265338393921}a^{25}-\frac{160}{265338393921}a^{24}-\frac{599}{265338393921}a^{23}-\frac{296837}{1594323}a^{22}-\frac{569878}{1594323}a^{21}-\frac{133934}{1594323}a^{20}+\frac{249245}{1594323}a^{19}+\frac{152557}{1594323}a^{18}+\frac{694031}{1594323}a^{17}+\frac{442621}{1594323}a^{16}+\frac{663932}{1594323}a^{15}-\frac{397472}{1594323}a^{14}-\frac{1}{1594323}a^{13}-\frac{16951}{88446131307}a^{12}+\frac{61126}{29482043769}a^{11}-\frac{14725}{9827347923}a^{10}-\frac{15467}{3275782641}a^{9}+\frac{10064}{1091927547}a^{8}+\frac{1801}{363975849}a^{7}-\frac{3955}{121325283}a^{6}+\frac{718}{40441761}a^{5}+\frac{1079}{13480587}a^{4}-\frac{599}{4493529}a^{3}-\frac{160}{1497843}a^{2}+\frac{253}{499281}a-\frac{31}{166427}$, $\frac{1}{796015181763}a^{36}-\frac{1}{796015181763}a^{35}-\frac{2}{796015181763}a^{34}+\frac{5}{796015181763}a^{33}+\frac{1}{796015181763}a^{32}-\frac{16}{796015181763}a^{31}+\frac{13}{796015181763}a^{30}+\frac{35}{796015181763}a^{29}-\frac{74}{796015181763}a^{28}-\frac{31}{796015181763}a^{27}+\frac{253}{796015181763}a^{26}-\frac{160}{796015181763}a^{25}-\frac{599}{796015181763}a^{24}+\frac{1079}{796015181763}a^{23}-\frac{2164201}{4782969}a^{22}-\frac{1728257}{4782969}a^{21}-\frac{1345078}{4782969}a^{20}+\frac{1746880}{4782969}a^{19}+\frac{2288354}{4782969}a^{18}+\frac{2036944}{4782969}a^{17}+\frac{663932}{4782969}a^{16}-\frac{1991795}{4782969}a^{15}-\frac{1}{4782969}a^{14}-\frac{16951}{265338393921}a^{13}+\frac{61126}{88446131307}a^{12}-\frac{14725}{29482043769}a^{11}-\frac{15467}{9827347923}a^{10}+\frac{10064}{3275782641}a^{9}+\frac{1801}{1091927547}a^{8}-\frac{3955}{363975849}a^{7}+\frac{718}{121325283}a^{6}+\frac{1079}{40441761}a^{5}-\frac{599}{13480587}a^{4}-\frac{160}{4493529}a^{3}+\frac{253}{1497843}a^{2}-\frac{31}{499281}a-\frac{74}{166427}$, $\frac{1}{2388045545289}a^{37}-\frac{1}{2388045545289}a^{36}-\frac{2}{2388045545289}a^{35}+\frac{5}{2388045545289}a^{34}+\frac{1}{2388045545289}a^{33}-\frac{16}{2388045545289}a^{32}+\frac{13}{2388045545289}a^{31}+\frac{35}{2388045545289}a^{30}-\frac{74}{2388045545289}a^{29}-\frac{31}{2388045545289}a^{28}+\frac{253}{2388045545289}a^{27}-\frac{160}{2388045545289}a^{26}-\frac{599}{2388045545289}a^{25}+\frac{1079}{2388045545289}a^{24}+\frac{718}{2388045545289}a^{23}-\frac{6511226}{14348907}a^{22}-\frac{1345078}{14348907}a^{21}+\frac{6529849}{14348907}a^{20}-\frac{2494615}{14348907}a^{19}-\frac{2746025}{14348907}a^{18}-\frac{4119037}{14348907}a^{17}-\frac{1991795}{14348907}a^{16}-\frac{1}{14348907}a^{15}-\frac{16951}{796015181763}a^{14}+\frac{61126}{265338393921}a^{13}-\frac{14725}{88446131307}a^{12}-\frac{15467}{29482043769}a^{11}+\frac{10064}{9827347923}a^{10}+\frac{1801}{3275782641}a^{9}-\frac{3955}{1091927547}a^{8}+\frac{718}{363975849}a^{7}+\frac{1079}{121325283}a^{6}-\frac{599}{40441761}a^{5}-\frac{160}{13480587}a^{4}+\frac{253}{4493529}a^{3}-\frac{31}{1497843}a^{2}-\frac{74}{499281}a+\frac{35}{166427}$, $\frac{1}{7164136635867}a^{38}-\frac{1}{7164136635867}a^{37}-\frac{2}{7164136635867}a^{36}+\frac{5}{7164136635867}a^{35}+\frac{1}{7164136635867}a^{34}-\frac{16}{7164136635867}a^{33}+\frac{13}{7164136635867}a^{32}+\frac{35}{7164136635867}a^{31}-\frac{74}{7164136635867}a^{30}-\frac{31}{7164136635867}a^{29}+\frac{253}{7164136635867}a^{28}-\frac{160}{7164136635867}a^{27}-\frac{599}{7164136635867}a^{26}+\frac{1079}{7164136635867}a^{25}+\frac{718}{7164136635867}a^{24}-\frac{3955}{7164136635867}a^{23}-\frac{15693985}{43046721}a^{22}-\frac{7819058}{43046721}a^{21}+\frac{11854292}{43046721}a^{20}+\frac{11602882}{43046721}a^{19}-\frac{4119037}{43046721}a^{18}+\frac{12357112}{43046721}a^{17}-\frac{1}{43046721}a^{16}-\frac{16951}{2388045545289}a^{15}+\frac{61126}{796015181763}a^{14}-\frac{14725}{265338393921}a^{13}-\frac{15467}{88446131307}a^{12}+\frac{10064}{29482043769}a^{11}+\frac{1801}{9827347923}a^{10}-\frac{3955}{3275782641}a^{9}+\frac{718}{1091927547}a^{8}+\frac{1079}{363975849}a^{7}-\frac{599}{121325283}a^{6}-\frac{160}{40441761}a^{5}+\frac{253}{13480587}a^{4}-\frac{31}{4493529}a^{3}-\frac{74}{1497843}a^{2}+\frac{35}{499281}a+\frac{13}{166427}$, $\frac{1}{21492409907601}a^{39}-\frac{1}{21492409907601}a^{38}-\frac{2}{21492409907601}a^{37}+\frac{5}{21492409907601}a^{36}+\frac{1}{21492409907601}a^{35}-\frac{16}{21492409907601}a^{34}+\frac{13}{21492409907601}a^{33}+\frac{35}{21492409907601}a^{32}-\frac{74}{21492409907601}a^{31}-\frac{31}{21492409907601}a^{30}+\frac{253}{21492409907601}a^{29}-\frac{160}{21492409907601}a^{28}-\frac{599}{21492409907601}a^{27}+\frac{1079}{21492409907601}a^{26}+\frac{718}{21492409907601}a^{25}-\frac{3955}{21492409907601}a^{24}+\frac{1801}{21492409907601}a^{23}+\frac{35227663}{129140163}a^{22}+\frac{11854292}{129140163}a^{21}+\frac{11602882}{129140163}a^{20}-\frac{47165758}{129140163}a^{19}+\frac{12357112}{129140163}a^{18}-\frac{1}{129140163}a^{17}-\frac{16951}{7164136635867}a^{16}+\frac{61126}{2388045545289}a^{15}-\frac{14725}{796015181763}a^{14}-\frac{15467}{265338393921}a^{13}+\frac{10064}{88446131307}a^{12}+\frac{1801}{29482043769}a^{11}-\frac{3955}{9827347923}a^{10}+\frac{718}{3275782641}a^{9}+\frac{1079}{1091927547}a^{8}-\frac{599}{363975849}a^{7}-\frac{160}{121325283}a^{6}+\frac{253}{40441761}a^{5}-\frac{31}{13480587}a^{4}-\frac{74}{4493529}a^{3}+\frac{35}{1497843}a^{2}+\frac{13}{499281}a-\frac{16}{166427}$, $\frac{1}{64477229722803}a^{40}-\frac{1}{64477229722803}a^{39}-\frac{2}{64477229722803}a^{38}+\frac{5}{64477229722803}a^{37}+\frac{1}{64477229722803}a^{36}-\frac{16}{64477229722803}a^{35}+\frac{13}{64477229722803}a^{34}+\frac{35}{64477229722803}a^{33}-\frac{74}{64477229722803}a^{32}-\frac{31}{64477229722803}a^{31}+\frac{253}{64477229722803}a^{30}-\frac{160}{64477229722803}a^{29}-\frac{599}{64477229722803}a^{28}+\frac{1079}{64477229722803}a^{27}+\frac{718}{64477229722803}a^{26}-\frac{3955}{64477229722803}a^{25}+\frac{1801}{64477229722803}a^{24}+\frac{10064}{64477229722803}a^{23}+\frac{140994455}{387420489}a^{22}+\frac{140743045}{387420489}a^{21}-\frac{176305921}{387420489}a^{20}+\frac{141497275}{387420489}a^{19}-\frac{1}{387420489}a^{18}-\frac{16951}{21492409907601}a^{17}+\frac{61126}{7164136635867}a^{16}-\frac{14725}{2388045545289}a^{15}-\frac{15467}{796015181763}a^{14}+\frac{10064}{265338393921}a^{13}+\frac{1801}{88446131307}a^{12}-\frac{3955}{29482043769}a^{11}+\frac{718}{9827347923}a^{10}+\frac{1079}{3275782641}a^{9}-\frac{599}{1091927547}a^{8}-\frac{160}{363975849}a^{7}+\frac{253}{121325283}a^{6}-\frac{31}{40441761}a^{5}-\frac{74}{13480587}a^{4}+\frac{35}{4493529}a^{3}+\frac{13}{1497843}a^{2}-\frac{16}{499281}a+\frac{1}{166427}$, $\frac{1}{193431689168409}a^{41}-\frac{1}{193431689168409}a^{40}-\frac{2}{193431689168409}a^{39}+\frac{5}{193431689168409}a^{38}+\frac{1}{193431689168409}a^{37}-\frac{16}{193431689168409}a^{36}+\frac{13}{193431689168409}a^{35}+\frac{35}{193431689168409}a^{34}-\frac{74}{193431689168409}a^{33}-\frac{31}{193431689168409}a^{32}+\frac{253}{193431689168409}a^{31}-\frac{160}{193431689168409}a^{30}-\frac{599}{193431689168409}a^{29}+\frac{1079}{193431689168409}a^{28}+\frac{718}{193431689168409}a^{27}-\frac{3955}{193431689168409}a^{26}+\frac{1801}{193431689168409}a^{25}+\frac{10064}{193431689168409}a^{24}-\frac{15467}{193431689168409}a^{23}+\frac{528163534}{1162261467}a^{22}+\frac{211114568}{1162261467}a^{21}+\frac{528917764}{1162261467}a^{20}-\frac{1}{1162261467}a^{19}-\frac{16951}{64477229722803}a^{18}+\frac{61126}{21492409907601}a^{17}-\frac{14725}{7164136635867}a^{16}-\frac{15467}{2388045545289}a^{15}+\frac{10064}{796015181763}a^{14}+\frac{1801}{265338393921}a^{13}-\frac{3955}{88446131307}a^{12}+\frac{718}{29482043769}a^{11}+\frac{1079}{9827347923}a^{10}-\frac{599}{3275782641}a^{9}-\frac{160}{1091927547}a^{8}+\frac{253}{363975849}a^{7}-\frac{31}{121325283}a^{6}-\frac{74}{40441761}a^{5}+\frac{35}{13480587}a^{4}+\frac{13}{4493529}a^{3}-\frac{16}{1497843}a^{2}+\frac{1}{499281}a+\frac{5}{166427}$, $\frac{1}{580295067505227}a^{42}-\frac{1}{580295067505227}a^{41}-\frac{2}{580295067505227}a^{40}+\frac{5}{580295067505227}a^{39}+\frac{1}{580295067505227}a^{38}-\frac{16}{580295067505227}a^{37}+\frac{13}{580295067505227}a^{36}+\frac{35}{580295067505227}a^{35}-\frac{74}{580295067505227}a^{34}-\frac{31}{580295067505227}a^{33}+\frac{253}{580295067505227}a^{32}-\frac{160}{580295067505227}a^{31}-\frac{599}{580295067505227}a^{30}+\frac{1079}{580295067505227}a^{29}+\frac{718}{580295067505227}a^{28}-\frac{3955}{580295067505227}a^{27}+\frac{1801}{580295067505227}a^{26}+\frac{10064}{580295067505227}a^{25}-\frac{15467}{580295067505227}a^{24}-\frac{14725}{580295067505227}a^{23}-\frac{951146899}{3486784401}a^{22}-\frac{633343703}{3486784401}a^{21}-\frac{1}{3486784401}a^{20}-\frac{16951}{193431689168409}a^{19}+\frac{61126}{64477229722803}a^{18}-\frac{14725}{21492409907601}a^{17}-\frac{15467}{7164136635867}a^{16}+\frac{10064}{2388045545289}a^{15}+\frac{1801}{796015181763}a^{14}-\frac{3955}{265338393921}a^{13}+\frac{718}{88446131307}a^{12}+\frac{1079}{29482043769}a^{11}-\frac{599}{9827347923}a^{10}-\frac{160}{3275782641}a^{9}+\frac{253}{1091927547}a^{8}-\frac{31}{363975849}a^{7}-\frac{74}{121325283}a^{6}+\frac{35}{40441761}a^{5}+\frac{13}{13480587}a^{4}-\frac{16}{4493529}a^{3}+\frac{1}{1497843}a^{2}+\frac{5}{499281}a-\frac{2}{166427}$, $\frac{1}{17\!\cdots\!81}a^{43}-\frac{1}{17\!\cdots\!81}a^{42}-\frac{2}{17\!\cdots\!81}a^{41}+\frac{5}{17\!\cdots\!81}a^{40}+\frac{1}{17\!\cdots\!81}a^{39}-\frac{16}{17\!\cdots\!81}a^{38}+\frac{13}{17\!\cdots\!81}a^{37}+\frac{35}{17\!\cdots\!81}a^{36}-\frac{74}{17\!\cdots\!81}a^{35}-\frac{31}{17\!\cdots\!81}a^{34}+\frac{253}{17\!\cdots\!81}a^{33}-\frac{160}{17\!\cdots\!81}a^{32}-\frac{599}{17\!\cdots\!81}a^{31}+\frac{1079}{17\!\cdots\!81}a^{30}+\frac{718}{17\!\cdots\!81}a^{29}-\frac{3955}{17\!\cdots\!81}a^{28}+\frac{1801}{17\!\cdots\!81}a^{27}+\frac{10064}{17\!\cdots\!81}a^{26}-\frac{15467}{17\!\cdots\!81}a^{25}-\frac{14725}{17\!\cdots\!81}a^{24}+\frac{61126}{17\!\cdots\!81}a^{23}+\frac{2853440698}{10460353203}a^{22}-\frac{1}{10460353203}a^{21}-\frac{16951}{580295067505227}a^{20}+\frac{61126}{193431689168409}a^{19}-\frac{14725}{64477229722803}a^{18}-\frac{15467}{21492409907601}a^{17}+\frac{10064}{7164136635867}a^{16}+\frac{1801}{2388045545289}a^{15}-\frac{3955}{796015181763}a^{14}+\frac{718}{265338393921}a^{13}+\frac{1079}{88446131307}a^{12}-\frac{599}{29482043769}a^{11}-\frac{160}{9827347923}a^{10}+\frac{253}{3275782641}a^{9}-\frac{31}{1091927547}a^{8}-\frac{74}{363975849}a^{7}+\frac{35}{121325283}a^{6}+\frac{13}{40441761}a^{5}-\frac{16}{13480587}a^{4}+\frac{1}{4493529}a^{3}+\frac{5}{1497843}a^{2}-\frac{2}{499281}a-\frac{1}{166427}$ Copy content Toggle raw display

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:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{15467}{193431689168409} a^{42} + \frac{1664094224}{193431689168409} a^{19} \)  (order $46$) Copy content Toggle raw display
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 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609)
 
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 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609, 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 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609);
 
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 - x^43 - 2*x^42 + 5*x^41 + x^40 - 16*x^39 + 13*x^38 + 35*x^37 - 74*x^36 - 31*x^35 + 253*x^34 - 160*x^33 - 599*x^32 + 1079*x^31 + 718*x^30 - 3955*x^29 + 1801*x^28 + 10064*x^27 - 15467*x^26 - 14725*x^25 + 61126*x^24 - 16951*x^23 - 166427*x^22 - 50853*x^21 + 550134*x^20 - 397575*x^19 - 1252827*x^18 + 2445552*x^17 + 1312929*x^16 - 8649585*x^15 + 4710798*x^14 + 21237957*x^13 - 35370351*x^12 - 28343520*x^11 + 134454573*x^10 - 49424013*x^9 - 353939706*x^8 + 502211745*x^7 + 559607373*x^6 - 2066242608*x^5 + 387420489*x^4 + 5811307335*x^3 - 6973568802*x^2 - 10460353203*x + 31381059609);
 
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 $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-11}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.489639287266880371204241478228795637811.1, 22.22.11261703607138248537697553999262299669653.1, \(\Q(\zeta_{23})\)

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 $22^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ $22^{2}$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display 23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$
23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$