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Properties

Label 44.0.361...241.1
Degree $44$
Signature $[0, 22]$
Discriminant $3.614\times 10^{83}$
Root discriminant \(79.26\)
Ramified primes $3,7,23$
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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304)
 
gp: K = bnfinit(y^44 - y^43 - 31*y^42 + 18*y^41 + 509*y^40 - 128*y^39 - 6323*y^38 + 711*y^37 + 64913*y^36 - 2855*y^35 - 548822*y^34 - 30450*y^33 + 3909507*y^32 + 497028*y^31 - 24053724*y^30 - 3501489*y^29 + 127000724*y^28 + 22049413*y^27 - 570173510*y^26 - 120800376*y^25 + 2191193723*y^24 + 441547756*y^23 - 7154579015*y^22 - 1235011630*y^21 + 19357719068*y^20 + 3692553824*y^19 - 42800540592*y^18 - 8560769600*y^17 + 77181753152*y^16 + 8780862080*y^15 - 109191557376*y^14 - 4336701952*y^13 + 115169783808*y^12 + 13857435648*y^11 - 86371995648*y^10 - 18961448960*y^9 + 47011856384*y^8 + 1711374336*y^7 - 14905901056*y^6 + 3448373248*y^5 + 3020947456*y^4 - 373293056*y^3 - 84934656*y^2 - 12582912*y + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304)
 

\( x^{44} - x^{43} - 31 x^{42} + 18 x^{41} + 509 x^{40} - 128 x^{39} - 6323 x^{38} + 711 x^{37} + \cdots + 4194304 \) 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:   \(361\!\cdots\!241\) \(\medspace = 3^{22}\cdot 7^{22}\cdot 23^{40}\) 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:  \(79.26\)
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:  $3^{1/2}7^{1/2}23^{10/11}\approx 79.2581705661848$
Ramified primes:   \(3\), \(7\), \(23\) 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:  \(483=3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{483}(1,·)$, $\chi_{483}(386,·)$, $\chi_{483}(8,·)$, $\chi_{483}(265,·)$, $\chi_{483}(139,·)$, $\chi_{483}(13,·)$, $\chi_{483}(400,·)$, $\chi_{483}(146,·)$, $\chi_{483}(407,·)$, $\chi_{483}(29,·)$, $\chi_{483}(167,·)$, $\chi_{483}(41,·)$, $\chi_{483}(302,·)$, $\chi_{483}(50,·)$, $\chi_{483}(307,·)$, $\chi_{483}(55,·)$, $\chi_{483}(440,·)$, $\chi_{483}(188,·)$, $\chi_{483}(190,·)$, $\chi_{483}(64,·)$, $\chi_{483}(449,·)$, $\chi_{483}(323,·)$, $\chi_{483}(197,·)$, $\chi_{483}(71,·)$, $\chi_{483}(328,·)$, $\chi_{483}(202,·)$, $\chi_{483}(461,·)$, $\chi_{483}(335,·)$, $\chi_{483}(209,·)$, $\chi_{483}(211,·)$, $\chi_{483}(85,·)$, $\chi_{483}(463,·)$, $\chi_{483}(349,·)$, $\chi_{483}(223,·)$, $\chi_{483}(358,·)$, $\chi_{483}(104,·)$, $\chi_{483}(239,·)$, $\chi_{483}(232,·)$, $\chi_{483}(370,·)$, $\chi_{483}(62,·)$, $\chi_{483}(118,·)$, $\chi_{483}(169,·)$, $\chi_{483}(377,·)$, $\chi_{483}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{24}-\frac{1}{4}a^{23}+\frac{1}{4}a^{22}-\frac{1}{2}a^{21}+\frac{1}{4}a^{20}+\frac{1}{4}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{25}-\frac{1}{8}a^{24}+\frac{1}{8}a^{23}+\frac{1}{4}a^{22}-\frac{3}{8}a^{21}-\frac{3}{8}a^{19}-\frac{1}{8}a^{18}+\frac{1}{8}a^{17}+\frac{1}{8}a^{16}+\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{3}{8}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{8}a^{10}-\frac{1}{2}a^{9}-\frac{3}{8}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{26}-\frac{1}{16}a^{25}+\frac{1}{16}a^{24}+\frac{1}{8}a^{23}-\frac{3}{16}a^{22}-\frac{3}{16}a^{20}+\frac{7}{16}a^{19}+\frac{1}{16}a^{18}-\frac{7}{16}a^{17}-\frac{3}{8}a^{16}-\frac{1}{8}a^{15}+\frac{3}{16}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{16}a^{11}+\frac{1}{4}a^{10}+\frac{5}{16}a^{9}-\frac{3}{8}a^{8}-\frac{1}{2}a^{7}-\frac{5}{16}a^{6}-\frac{1}{4}a^{5}-\frac{7}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{27}-\frac{1}{32}a^{26}+\frac{1}{32}a^{25}+\frac{1}{16}a^{24}-\frac{3}{32}a^{23}-\frac{3}{32}a^{21}-\frac{9}{32}a^{20}+\frac{1}{32}a^{19}+\frac{9}{32}a^{18}+\frac{5}{16}a^{17}+\frac{7}{16}a^{16}+\frac{3}{32}a^{15}-\frac{3}{8}a^{14}+\frac{1}{8}a^{13}-\frac{1}{32}a^{12}+\frac{1}{8}a^{11}+\frac{5}{32}a^{10}+\frac{5}{16}a^{9}-\frac{1}{4}a^{8}+\frac{11}{32}a^{7}-\frac{1}{8}a^{6}-\frac{7}{32}a^{5}+\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{28}-\frac{1}{64}a^{27}+\frac{1}{64}a^{26}+\frac{1}{32}a^{25}-\frac{3}{64}a^{24}+\frac{29}{64}a^{22}+\frac{23}{64}a^{21}-\frac{31}{64}a^{20}-\frac{23}{64}a^{19}-\frac{11}{32}a^{18}-\frac{9}{32}a^{17}-\frac{29}{64}a^{16}-\frac{3}{16}a^{15}-\frac{7}{16}a^{14}+\frac{31}{64}a^{13}-\frac{7}{16}a^{12}+\frac{5}{64}a^{11}-\frac{11}{32}a^{10}-\frac{1}{8}a^{9}-\frac{21}{64}a^{8}-\frac{1}{16}a^{7}+\frac{25}{64}a^{6}+\frac{1}{32}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{128}a^{29}-\frac{1}{128}a^{28}+\frac{1}{128}a^{27}+\frac{1}{64}a^{26}-\frac{3}{128}a^{25}+\frac{29}{128}a^{23}-\frac{41}{128}a^{22}-\frac{31}{128}a^{21}+\frac{41}{128}a^{20}+\frac{21}{64}a^{19}-\frac{9}{64}a^{18}-\frac{29}{128}a^{17}+\frac{13}{32}a^{16}+\frac{9}{32}a^{15}+\frac{31}{128}a^{14}-\frac{7}{32}a^{13}-\frac{59}{128}a^{12}+\frac{21}{64}a^{11}-\frac{1}{16}a^{10}+\frac{43}{128}a^{9}-\frac{1}{32}a^{8}+\frac{25}{128}a^{7}+\frac{1}{64}a^{6}-\frac{13}{32}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{256}a^{30}-\frac{1}{256}a^{29}+\frac{1}{256}a^{28}+\frac{1}{128}a^{27}-\frac{3}{256}a^{26}+\frac{29}{256}a^{24}-\frac{41}{256}a^{23}+\frac{97}{256}a^{22}+\frac{41}{256}a^{21}+\frac{21}{128}a^{20}-\frac{9}{128}a^{19}+\frac{99}{256}a^{18}-\frac{19}{64}a^{17}-\frac{23}{64}a^{16}+\frac{31}{256}a^{15}-\frac{7}{64}a^{14}+\frac{69}{256}a^{13}+\frac{21}{128}a^{12}+\frac{15}{32}a^{11}-\frac{85}{256}a^{10}+\frac{31}{64}a^{9}+\frac{25}{256}a^{8}+\frac{1}{128}a^{7}+\frac{19}{64}a^{6}-\frac{3}{16}a^{5}-\frac{3}{8}a^{4}$, $\frac{1}{512}a^{31}-\frac{1}{512}a^{30}+\frac{1}{512}a^{29}+\frac{1}{256}a^{28}-\frac{3}{512}a^{27}+\frac{29}{512}a^{25}-\frac{41}{512}a^{24}+\frac{97}{512}a^{23}+\frac{41}{512}a^{22}-\frac{107}{256}a^{21}+\frac{119}{256}a^{20}+\frac{99}{512}a^{19}+\frac{45}{128}a^{18}+\frac{41}{128}a^{17}-\frac{225}{512}a^{16}+\frac{57}{128}a^{15}+\frac{69}{512}a^{14}-\frac{107}{256}a^{13}-\frac{17}{64}a^{12}+\frac{171}{512}a^{11}+\frac{31}{128}a^{10}-\frac{231}{512}a^{9}+\frac{1}{256}a^{8}+\frac{19}{128}a^{7}+\frac{13}{32}a^{6}-\frac{3}{16}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{32}-\frac{1}{1024}a^{31}+\frac{1}{1024}a^{30}+\frac{1}{512}a^{29}-\frac{3}{1024}a^{28}+\frac{29}{1024}a^{26}-\frac{41}{1024}a^{25}+\frac{97}{1024}a^{24}+\frac{41}{1024}a^{23}+\frac{149}{512}a^{22}-\frac{137}{512}a^{21}+\frac{99}{1024}a^{20}+\frac{45}{256}a^{19}+\frac{41}{256}a^{18}-\frac{225}{1024}a^{17}+\frac{57}{256}a^{16}+\frac{69}{1024}a^{15}-\frac{107}{512}a^{14}+\frac{47}{128}a^{13}-\frac{341}{1024}a^{12}-\frac{97}{256}a^{11}-\frac{231}{1024}a^{10}+\frac{1}{512}a^{9}+\frac{19}{256}a^{8}-\frac{19}{64}a^{7}+\frac{13}{32}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{33}-\frac{1}{2048}a^{32}+\frac{1}{2048}a^{31}+\frac{1}{1024}a^{30}-\frac{3}{2048}a^{29}+\frac{29}{2048}a^{27}-\frac{41}{2048}a^{26}+\frac{97}{2048}a^{25}+\frac{41}{2048}a^{24}+\frac{149}{1024}a^{23}-\frac{137}{1024}a^{22}-\frac{925}{2048}a^{21}-\frac{211}{512}a^{20}+\frac{41}{512}a^{19}-\frac{225}{2048}a^{18}+\frac{57}{512}a^{17}-\frac{955}{2048}a^{16}-\frac{107}{1024}a^{15}+\frac{47}{256}a^{14}+\frac{683}{2048}a^{13}+\frac{159}{512}a^{12}-\frac{231}{2048}a^{11}-\frac{511}{1024}a^{10}-\frac{237}{512}a^{9}+\frac{45}{128}a^{8}-\frac{19}{64}a^{7}+\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4096}a^{34}-\frac{1}{4096}a^{33}+\frac{1}{4096}a^{32}+\frac{1}{2048}a^{31}-\frac{3}{4096}a^{30}+\frac{29}{4096}a^{28}-\frac{41}{4096}a^{27}+\frac{97}{4096}a^{26}+\frac{41}{4096}a^{25}+\frac{149}{2048}a^{24}-\frac{137}{2048}a^{23}-\frac{925}{4096}a^{22}+\frac{301}{1024}a^{21}+\frac{41}{1024}a^{20}+\frac{1823}{4096}a^{19}+\frac{57}{1024}a^{18}-\frac{955}{4096}a^{17}+\frac{917}{2048}a^{16}+\frac{47}{512}a^{15}-\frac{1365}{4096}a^{14}+\frac{159}{1024}a^{13}+\frac{1817}{4096}a^{12}-\frac{511}{2048}a^{11}-\frac{237}{1024}a^{10}+\frac{45}{256}a^{9}+\frac{45}{128}a^{8}-\frac{3}{8}a^{7}+\frac{5}{16}a^{6}-\frac{3}{8}a^{5}-\frac{1}{16}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8192}a^{35}-\frac{1}{8192}a^{34}+\frac{1}{8192}a^{33}+\frac{1}{4096}a^{32}-\frac{3}{8192}a^{31}+\frac{29}{8192}a^{29}-\frac{41}{8192}a^{28}+\frac{97}{8192}a^{27}+\frac{41}{8192}a^{26}+\frac{149}{4096}a^{25}-\frac{137}{4096}a^{24}-\frac{925}{8192}a^{23}+\frac{301}{2048}a^{22}-\frac{983}{2048}a^{21}+\frac{1823}{8192}a^{20}+\frac{57}{2048}a^{19}-\frac{955}{8192}a^{18}+\frac{917}{4096}a^{17}+\frac{47}{1024}a^{16}-\frac{1365}{8192}a^{15}-\frac{865}{2048}a^{14}-\frac{2279}{8192}a^{13}+\frac{1537}{4096}a^{12}-\frac{237}{2048}a^{11}+\frac{45}{512}a^{10}+\frac{45}{256}a^{9}-\frac{3}{16}a^{8}-\frac{11}{32}a^{7}-\frac{3}{16}a^{6}-\frac{1}{32}a^{5}+\frac{7}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16384}a^{36}-\frac{1}{16384}a^{35}+\frac{1}{16384}a^{34}+\frac{1}{8192}a^{33}-\frac{3}{16384}a^{32}+\frac{29}{16384}a^{30}-\frac{41}{16384}a^{29}+\frac{97}{16384}a^{28}+\frac{41}{16384}a^{27}+\frac{149}{8192}a^{26}-\frac{137}{8192}a^{25}-\frac{925}{16384}a^{24}+\frac{301}{4096}a^{23}-\frac{983}{4096}a^{22}+\frac{1823}{16384}a^{21}+\frac{57}{4096}a^{20}+\frac{7237}{16384}a^{19}-\frac{3179}{8192}a^{18}-\frac{977}{2048}a^{17}+\frac{6827}{16384}a^{16}-\frac{865}{4096}a^{15}-\frac{2279}{16384}a^{14}-\frac{2559}{8192}a^{13}+\frac{1811}{4096}a^{12}-\frac{467}{1024}a^{11}-\frac{211}{512}a^{10}+\frac{13}{32}a^{9}-\frac{11}{64}a^{8}+\frac{13}{32}a^{7}+\frac{31}{64}a^{6}+\frac{7}{32}a^{5}+\frac{1}{16}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32768}a^{37}-\frac{1}{32768}a^{36}+\frac{1}{32768}a^{35}+\frac{1}{16384}a^{34}-\frac{3}{32768}a^{33}+\frac{29}{32768}a^{31}-\frac{41}{32768}a^{30}+\frac{97}{32768}a^{29}+\frac{41}{32768}a^{28}+\frac{149}{16384}a^{27}-\frac{137}{16384}a^{26}-\frac{925}{32768}a^{25}+\frac{301}{8192}a^{24}-\frac{983}{8192}a^{23}+\frac{1823}{32768}a^{22}-\frac{4039}{8192}a^{21}+\frac{7237}{32768}a^{20}+\frac{5013}{16384}a^{19}-\frac{977}{4096}a^{18}+\frac{6827}{32768}a^{17}-\frac{865}{8192}a^{16}+\frac{14105}{32768}a^{15}+\frac{5633}{16384}a^{14}+\frac{1811}{8192}a^{13}+\frac{557}{2048}a^{12}-\frac{211}{1024}a^{11}+\frac{13}{64}a^{10}+\frac{53}{128}a^{9}+\frac{13}{64}a^{8}-\frac{33}{128}a^{7}+\frac{7}{64}a^{6}+\frac{1}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{65536}a^{38}-\frac{1}{65536}a^{37}+\frac{1}{65536}a^{36}+\frac{1}{32768}a^{35}-\frac{3}{65536}a^{34}+\frac{29}{65536}a^{32}-\frac{41}{65536}a^{31}+\frac{97}{65536}a^{30}+\frac{41}{65536}a^{29}+\frac{149}{32768}a^{28}-\frac{137}{32768}a^{27}-\frac{925}{65536}a^{26}+\frac{301}{16384}a^{25}-\frac{983}{16384}a^{24}+\frac{1823}{65536}a^{23}-\frac{4039}{16384}a^{22}+\frac{7237}{65536}a^{21}-\frac{11371}{32768}a^{20}-\frac{977}{8192}a^{19}-\frac{25941}{65536}a^{18}-\frac{865}{16384}a^{17}+\frac{14105}{65536}a^{16}-\frac{10751}{32768}a^{15}-\frac{6381}{16384}a^{14}+\frac{557}{4096}a^{13}-\frac{211}{2048}a^{12}+\frac{13}{128}a^{11}-\frac{75}{256}a^{10}-\frac{51}{128}a^{9}+\frac{95}{256}a^{8}-\frac{57}{128}a^{7}+\frac{1}{64}a^{6}+\frac{7}{16}a^{5}+\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{131072}a^{39}-\frac{1}{131072}a^{38}+\frac{1}{131072}a^{37}+\frac{1}{65536}a^{36}-\frac{3}{131072}a^{35}+\frac{29}{131072}a^{33}-\frac{41}{131072}a^{32}+\frac{97}{131072}a^{31}+\frac{41}{131072}a^{30}+\frac{149}{65536}a^{29}-\frac{137}{65536}a^{28}-\frac{925}{131072}a^{27}+\frac{301}{32768}a^{26}-\frac{983}{32768}a^{25}+\frac{1823}{131072}a^{24}-\frac{4039}{32768}a^{23}+\frac{7237}{131072}a^{22}-\frac{11371}{65536}a^{21}-\frac{977}{16384}a^{20}-\frac{25941}{131072}a^{19}+\frac{15519}{32768}a^{18}-\frac{51431}{131072}a^{17}+\frac{22017}{65536}a^{16}-\frac{6381}{32768}a^{15}-\frac{3539}{8192}a^{14}-\frac{211}{4096}a^{13}-\frac{115}{256}a^{12}+\frac{181}{512}a^{11}-\frac{51}{256}a^{10}-\frac{161}{512}a^{9}+\frac{71}{256}a^{8}-\frac{63}{128}a^{7}-\frac{9}{32}a^{6}-\frac{15}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{262144}a^{40}-\frac{1}{262144}a^{39}+\frac{1}{262144}a^{38}+\frac{1}{131072}a^{37}-\frac{3}{262144}a^{36}+\frac{29}{262144}a^{34}-\frac{41}{262144}a^{33}+\frac{97}{262144}a^{32}+\frac{41}{262144}a^{31}+\frac{149}{131072}a^{30}-\frac{137}{131072}a^{29}-\frac{925}{262144}a^{28}+\frac{301}{65536}a^{27}-\frac{983}{65536}a^{26}+\frac{1823}{262144}a^{25}-\frac{4039}{65536}a^{24}+\frac{7237}{262144}a^{23}+\frac{54165}{131072}a^{22}+\frac{15407}{32768}a^{21}-\frac{25941}{262144}a^{20}-\frac{17249}{65536}a^{19}-\frac{51431}{262144}a^{18}-\frac{43519}{131072}a^{17}+\frac{26387}{65536}a^{16}+\frac{4653}{16384}a^{15}+\frac{3885}{8192}a^{14}-\frac{115}{512}a^{13}-\frac{331}{1024}a^{12}-\frac{51}{512}a^{11}-\frac{161}{1024}a^{10}-\frac{185}{512}a^{9}-\frac{63}{256}a^{8}-\frac{9}{64}a^{7}-\frac{15}{64}a^{6}-\frac{5}{32}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{145227776}a^{41}+\frac{11}{145227776}a^{40}-\frac{259}{145227776}a^{39}-\frac{545}{72613888}a^{38}-\frac{27}{145227776}a^{37}-\frac{951}{36306944}a^{36}+\frac{7413}{145227776}a^{35}-\frac{13685}{145227776}a^{34}+\frac{13149}{145227776}a^{33}-\frac{47419}{145227776}a^{32}-\frac{877}{72613888}a^{31}-\frac{94349}{72613888}a^{30}-\frac{145197}{145227776}a^{29}+\frac{71327}{18153472}a^{28}+\frac{138671}{36306944}a^{27}+\frac{71959}{145227776}a^{26}-\frac{507301}{18153472}a^{25}-\frac{5746739}{145227776}a^{24}+\frac{6171623}{72613888}a^{23}-\frac{4591889}{18153472}a^{22}-\frac{66852557}{145227776}a^{21}+\frac{2863599}{9076736}a^{20}-\frac{20534015}{145227776}a^{19}+\frac{10703227}{72613888}a^{18}-\frac{2267437}{36306944}a^{17}+\frac{5027693}{18153472}a^{16}-\frac{4060197}{9076736}a^{15}+\frac{1488833}{4538368}a^{14}-\frac{46931}{1134592}a^{13}+\frac{411149}{1134592}a^{12}+\frac{65421}{141824}a^{11}+\frac{20795}{141824}a^{10}+\frac{51437}{141824}a^{9}+\frac{22293}{70912}a^{8}+\frac{1457}{8864}a^{7}+\frac{97}{17728}a^{6}+\frac{1107}{4432}a^{5}-\frac{1227}{4432}a^{4}-\frac{41}{2216}a^{3}-\frac{225}{554}a^{2}-\frac{243}{554}a+\frac{29}{277}$, $\frac{1}{10\!\cdots\!12}a^{42}-\frac{13\!\cdots\!13}{10\!\cdots\!12}a^{41}+\frac{69\!\cdots\!09}{10\!\cdots\!12}a^{40}+\frac{17\!\cdots\!71}{51\!\cdots\!56}a^{39}+\frac{54\!\cdots\!29}{10\!\cdots\!12}a^{38}+\frac{14\!\cdots\!31}{25\!\cdots\!28}a^{37}-\frac{15\!\cdots\!35}{10\!\cdots\!12}a^{36}-\frac{44\!\cdots\!87}{16\!\cdots\!84}a^{35}-\frac{31\!\cdots\!43}{10\!\cdots\!12}a^{34}-\frac{22\!\cdots\!15}{10\!\cdots\!12}a^{33}-\frac{63\!\cdots\!21}{51\!\cdots\!56}a^{32}-\frac{16\!\cdots\!41}{51\!\cdots\!56}a^{31}+\frac{96\!\cdots\!15}{10\!\cdots\!12}a^{30}+\frac{38\!\cdots\!29}{12\!\cdots\!64}a^{29}-\frac{88\!\cdots\!23}{25\!\cdots\!28}a^{28}-\frac{43\!\cdots\!49}{10\!\cdots\!12}a^{27}-\frac{15\!\cdots\!13}{10\!\cdots\!44}a^{26}+\frac{53\!\cdots\!77}{10\!\cdots\!12}a^{25}-\frac{42\!\cdots\!17}{51\!\cdots\!56}a^{24}-\frac{84\!\cdots\!25}{64\!\cdots\!32}a^{23}+\frac{25\!\cdots\!11}{75\!\cdots\!76}a^{22}-\frac{13\!\cdots\!33}{64\!\cdots\!32}a^{21}+\frac{18\!\cdots\!85}{10\!\cdots\!12}a^{20}-\frac{49\!\cdots\!09}{51\!\cdots\!56}a^{19}-\frac{11\!\cdots\!51}{25\!\cdots\!28}a^{18}+\frac{74\!\cdots\!49}{16\!\cdots\!08}a^{17}-\frac{15\!\cdots\!99}{32\!\cdots\!16}a^{16}+\frac{14\!\cdots\!43}{32\!\cdots\!16}a^{15}-\frac{40\!\cdots\!55}{20\!\cdots\!76}a^{14}-\frac{22\!\cdots\!39}{80\!\cdots\!04}a^{13}-\frac{68\!\cdots\!03}{40\!\cdots\!52}a^{12}-\frac{49\!\cdots\!91}{10\!\cdots\!88}a^{11}+\frac{58\!\cdots\!87}{12\!\cdots\!36}a^{10}+\frac{15\!\cdots\!57}{50\!\cdots\!44}a^{9}-\frac{77\!\cdots\!31}{25\!\cdots\!72}a^{8}+\frac{20\!\cdots\!37}{12\!\cdots\!36}a^{7}-\frac{43\!\cdots\!25}{15\!\cdots\!92}a^{6}+\frac{67\!\cdots\!43}{31\!\cdots\!84}a^{5}+\frac{19\!\cdots\!61}{15\!\cdots\!92}a^{4}+\frac{29\!\cdots\!17}{78\!\cdots\!96}a^{3}-\frac{25\!\cdots\!21}{98\!\cdots\!37}a^{2}-\frac{93\!\cdots\!10}{98\!\cdots\!37}a+\frac{23\!\cdots\!04}{98\!\cdots\!37}$, $\frac{1}{47\!\cdots\!16}a^{43}+\frac{25\!\cdots\!81}{47\!\cdots\!16}a^{42}-\frac{26\!\cdots\!17}{47\!\cdots\!16}a^{41}-\frac{27\!\cdots\!61}{29\!\cdots\!76}a^{40}+\frac{11\!\cdots\!61}{47\!\cdots\!16}a^{39}+\frac{56\!\cdots\!73}{23\!\cdots\!08}a^{38}-\frac{36\!\cdots\!07}{47\!\cdots\!16}a^{37}-\frac{10\!\cdots\!27}{47\!\cdots\!16}a^{36}+\frac{27\!\cdots\!47}{47\!\cdots\!16}a^{35}-\frac{18\!\cdots\!97}{47\!\cdots\!16}a^{34}+\frac{25\!\cdots\!55}{29\!\cdots\!76}a^{33}+\frac{73\!\cdots\!35}{23\!\cdots\!08}a^{32}-\frac{26\!\cdots\!77}{47\!\cdots\!16}a^{31}-\frac{15\!\cdots\!27}{23\!\cdots\!08}a^{30}-\frac{75\!\cdots\!93}{29\!\cdots\!76}a^{29}+\frac{27\!\cdots\!71}{47\!\cdots\!16}a^{28}-\frac{30\!\cdots\!15}{23\!\cdots\!08}a^{27}-\frac{48\!\cdots\!91}{47\!\cdots\!16}a^{26}+\frac{10\!\cdots\!11}{11\!\cdots\!04}a^{25}+\frac{11\!\cdots\!01}{59\!\cdots\!52}a^{24}-\frac{21\!\cdots\!77}{47\!\cdots\!16}a^{23}-\frac{49\!\cdots\!55}{23\!\cdots\!08}a^{22}+\frac{17\!\cdots\!13}{47\!\cdots\!16}a^{21}-\frac{42\!\cdots\!69}{11\!\cdots\!04}a^{20}+\frac{52\!\cdots\!47}{11\!\cdots\!04}a^{19}+\frac{41\!\cdots\!75}{29\!\cdots\!76}a^{18}+\frac{67\!\cdots\!89}{37\!\cdots\!72}a^{17}+\frac{52\!\cdots\!39}{14\!\cdots\!88}a^{16}-\frac{15\!\cdots\!09}{74\!\cdots\!44}a^{15}+\frac{72\!\cdots\!65}{37\!\cdots\!72}a^{14}-\frac{12\!\cdots\!89}{93\!\cdots\!68}a^{13}-\frac{13\!\cdots\!89}{93\!\cdots\!68}a^{12}-\frac{21\!\cdots\!07}{46\!\cdots\!84}a^{11}+\frac{13\!\cdots\!27}{58\!\cdots\!48}a^{10}+\frac{39\!\cdots\!57}{11\!\cdots\!96}a^{9}+\frac{66\!\cdots\!49}{58\!\cdots\!48}a^{8}+\frac{68\!\cdots\!53}{14\!\cdots\!12}a^{7}+\frac{22\!\cdots\!05}{14\!\cdots\!12}a^{6}-\frac{34\!\cdots\!29}{72\!\cdots\!56}a^{5}+\frac{14\!\cdots\!55}{36\!\cdots\!28}a^{4}-\frac{13\!\cdots\!96}{22\!\cdots\!83}a^{3}-\frac{24\!\cdots\!77}{90\!\cdots\!32}a^{2}-\frac{30\!\cdots\!41}{22\!\cdots\!83}a-\frac{27\!\cdots\!68}{22\!\cdots\!83}$ 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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{59838665153067160308102478878484539824480056922996626813673049472303180365121279112484782633535721185}{1006351277064325307064667407457681010366404206095502051237714880270238652718031429111830593120898374879870976} a^{43} + \frac{651470009122725157247350178378432883282436014307628080387437333425456189220329283080297624969180739541}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{42} + \frac{29958835976203850355238480119536536945427696205701793084693641906215186332231013714585536080744082885555}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{41} - \frac{7741339186307442958396550314303865852571863007490090387198049670650103652248267109921690033996826290727}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{40} - \frac{245986830561550650931370421947024640066333055315634808461124729512941854790992210586902482711193669572469}{8050810216514602456517339259661448082931233648764016409901719042161909221744251432894644744967186999038967808} a^{39} - \frac{33073628065961557686547307010443844140777950894174945825928731302195422108131547724483477786805324314659}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{38} + \frac{379923528047704466794658213125334947887978584759056195028084342738804234931770127470447906359037056071729}{1006351277064325307064667407457681010366404206095502051237714880270238652718031429111830593120898374879870976} a^{37} + \frac{1247611840649057471512668824067395581905107879989564843343187155868721378177380313648729579760847965407709}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{36} - \frac{62192580642797943491972384127862228262346530046178231783711125979024183896539636518684794234073003727508517}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{35} - \frac{17026679736589819781015605776436126563390670637098005720927912606350698186323631581155940758727747913670647}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{34} + \frac{524551839518729717455198610366902506401961656091111370117843196698347210493803511147150635916587171117992585}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{33} + \frac{97961928651767511335190330948432400077061265931926075677862296723230035244579073972112301867019765595981195}{8050810216514602456517339259661448082931233648764016409901719042161909221744251432894644744967186999038967808} a^{32} - \frac{1859387031515792759389053349860325573005787176706820013631917063724165118548972678113492872775367541946396879}{8050810216514602456517339259661448082931233648764016409901719042161909221744251432894644744967186999038967808} a^{31} - \frac{1660905670127505826108209825320031290098413568969020845759619048082995930200418944498888782503370410458135209}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{30} + \frac{5692974077435418532628892105004647620617221694141165880069787814022023242507506943731615169460485983915195301}{4025405108257301228258669629830724041465616824382008204950859521080954610872125716447322372483593499519483904} a^{29} + \frac{1328198712412357928891369456786614036563813860813694111812654029413837801951832625505271591942974552468959959}{2012702554128650614129334814915362020732808412191004102475429760540477305436062858223661186241796749759741952} a^{28} - \frac{119877854000314062669197534586026076256345823929065735905629316477039263898723392562386561403734037596034881285}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{27} - \frac{14856886559211176463417727597137076386544670584027641098706894752611544684328344736879559285663997325293823977}{4025405108257301228258669629830724041465616824382008204950859521080954610872125716447322372483593499519483904} a^{26} + \frac{535763833102983933552271566938744363265422486383340949697842084010981634814207447520624565867085865972116009085}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{25} + \frac{143588050090245975013791934994349222707716979166681357081104902193656997931896233310512235838929492534721863181}{8050810216514602456517339259661448082931233648764016409901719042161909221744251432894644744967186999038967808} a^{24} - \frac{511460817442399027241420379850820290550087377065327816701530237676948139567276922782498341650529121297059781667}{4025405108257301228258669629830724041465616824382008204950859521080954610872125716447322372483593499519483904} a^{23} - \frac{2941438964681803633749003103454691975066359662481063578231457572099049816195645088368111253125214685663789871}{43873625158117724558677598145293995002350047132228972261044790420500867693429163122041660735516005444354048} a^{22} + \frac{1664368959074039639483888947312203921233742436120401032189706349293114103784079562839027751179132273860306796297}{4025405108257301228258669629830724041465616824382008204950859521080954610872125716447322372483593499519483904} a^{21} + \frac{3319953336050242347893975086616992702299767275481496316493817531430344519237868981456507585069536443065758690129}{16101620433029204913034678519322896165862467297528032819803438084323818443488502865789289489934373998077935616} a^{20} - \frac{8984423850939982113062009276608922889159702605482605182612437760986792122662693937555677789129510184032783677047}{8050810216514602456517339259661448082931233648764016409901719042161909221744251432894644744967186999038967808} a^{19} - \frac{1162128019247399626921858826117898370905295073398737694230520378074335947747461030774201769349924123800123076993}{2012702554128650614129334814915362020732808412191004102475429760540477305436062858223661186241796749759741952} a^{18} + \frac{4917865188218440591053943686002765218621008248916287662140234879914142012124300734710369037181801814076402752459}{2012702554128650614129334814915362020732808412191004102475429760540477305436062858223661186241796749759741952} a^{17} + \frac{40726464536546734100010892719205881580991107959648299209252972584605545017349288025516622893546593759943128933}{31448477408260165845770856483052531573950131440484439101178590008444957897438482159744706035028074214995968} a^{16} - \frac{1096484985443518501688486317341035231595797499679871858671401410063888180135767098808074742605041568207867764509}{251587819266081326766166851864420252591601051523875512809428720067559663179507857277957648280224593719967744} a^{15} - \frac{121302981166394235601334722441505081417517036613921700563617031235404321146592146969826662748957253212363077743}{62896954816520331691541712966105063147900262880968878202357180016889915794876964319489412070056148429991936} a^{14} + \frac{780446453962642854979362618367097835240939311481010188777240108773044754748432744916855193560535882234251426087}{125793909633040663383083425932210126295800525761937756404714360033779831589753928638978824140112296859983872} a^{13} + \frac{70363025880705449010588767253618094863797762418966104126587422058437411355405039778066519092164290402445329585}{31448477408260165845770856483052531573950131440484439101178590008444957897438482159744706035028074214995968} a^{12} - \frac{207198891152168512315981875581609481631837243823446010019320527065340552104513557415986838732750933098896217417}{31448477408260165845770856483052531573950131440484439101178590008444957897438482159744706035028074214995968} a^{11} - \frac{45526249985803258317824767218367820664795004704832403424481455057014193832543102394439795747859216331970544117}{15724238704130082922885428241526265786975065720242219550589295004222478948719241079872353017514037107497984} a^{10} + \frac{36794525126079605772830083621298273582194153761487415077516938711881275347405583900377296923937550920948888547}{7862119352065041461442714120763132893487532860121109775294647502111239474359620539936176508757018553748992} a^{9} + \frac{10361329822635635552078149775002640945903296885740447791132213828613147426218240423559217697043677848481149707}{3931059676032520730721357060381566446743766430060554887647323751055619737179810269968088254378509276874496} a^{8} - \frac{2264121517767584691428690777457925714608529730651525921570061247256203448196429859427815574471779669328154519}{982764919008130182680339265095391611685941607515138721911830937763904934294952567492022063594627319218624} a^{7} - \frac{871767240630232873163226505271728710711627523367500454601010585092155940656128105935306375037213961689309577}{982764919008130182680339265095391611685941607515138721911830937763904934294952567492022063594627319218624} a^{6} + \frac{389194039147362857525680194762640234175145339498090566777013938869200508183991090359645107719678528909696723}{491382459504065091340169632547695805842970803757569360955915468881952467147476283746011031797313659609312} a^{5} + \frac{4571276639754770089152055695847707822664352390497028187589958546888045697375222538750827155916689441432151}{122845614876016272835042408136923951460742700939392340238978867220488116786869070936502757949328414902328} a^{4} - \frac{6684898508994028885202491294986170943122651828539284227252017862249879083078178690750581814286317227480707}{30711403719004068208760602034230987865185675234848085059744716805122029196717267734125689487332103725582} a^{3} - \frac{2035685747753984286004199298598514010253725515859738686198585955428439367389699467882154798706499435563529}{61422807438008136417521204068461975730371350469696170119489433610244058393434535468251378974664207451164} a^{2} + \frac{125183976851136400012965776548819870282519490676377800594681232065737711932320785910276899753828182957211}{30711403719004068208760602034230987865185675234848085059744716805122029196717267734125689487332103725582} a + \frac{32616111369500136765372718269021861971469753006145111097021476736707178163301599645570112088629862023082}{15355701859502034104380301017115493932592837617424042529872358402561014598358633867062844743666051862791} \)  (order $6$) 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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304)
 
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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304, 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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304);
 
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 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304);
 
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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{23})^+\), 22.22.601130775140836298755595442714814879781421.1, 22.0.304011857053427966889939263171547.1, 22.0.3393400820453274956705986794666660343.1

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 $22^{2}$ R $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ ${\href{/padicField/37.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(7\) Copy content Toggle raw display 7.22.11.2$x^{22} + 77 x^{20} + 2695 x^{18} + 56595 x^{16} + 792330 x^{14} + 7764836 x^{12} + 8 x^{11} + 54353222 x^{10} - 3080 x^{9} + 271785360 x^{8} + 129360 x^{7} + 951192165 x^{6} - 1267728 x^{5} + 2218656055 x^{4} + 3169320 x^{3} + 3108706756 x^{2} - 1479008 x + 1977091468$$2$$11$$11$22T1$[\ ]_{2}^{11}$
7.22.11.2$x^{22} + 77 x^{20} + 2695 x^{18} + 56595 x^{16} + 792330 x^{14} + 7764836 x^{12} + 8 x^{11} + 54353222 x^{10} - 3080 x^{9} + 271785360 x^{8} + 129360 x^{7} + 951192165 x^{6} - 1267728 x^{5} + 2218656055 x^{4} + 3169320 x^{3} + 3108706756 x^{2} - 1479008 x + 1977091468$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(23\) Copy content Toggle raw display 23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$
23.22.20.1$x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$$11$$2$$20$22T1$[\ ]_{11}^{2}$