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Properties

Label 44.0.202...984.1
Degree $44$
Signature $[0, 22]$
Discriminant $2.026\times 10^{86}$
Root discriminant \(91.52\)
Ramified primes $2,7,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 4194304)
 
gp: K = bnfinit(y^44 - 63*y^42 + 1954*y^40 - 39159*y^38 + 564136*y^36 - 6171204*y^34 + 52905976*y^32 - 362220741*y^30 + 2002105655*y^28 - 8980281297*y^26 + 32710198942*y^24 - 96439370415*y^22 + 228524812948*y^20 - 430395212448*y^18 + 634098522688*y^16 - 714985491456*y^14 + 598979244032*y^12 - 357650202624*y^10 + 143735308288*y^8 - 35340484608*y^6 + 4755030016*y^4 - 207618048*y^2 + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 4194304);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 4194304)
 

\( x^{44} - 63 x^{42} + 1954 x^{40} - 39159 x^{38} + 564136 x^{36} - 6171204 x^{34} + 52905976 x^{32} + \cdots + 4194304 \) 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:   \(202\!\cdots\!984\) \(\medspace = 2^{44}\cdot 7^{22}\cdot 23^{40}\) 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:  \(91.52\)
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:  $2\cdot 7^{1/2}23^{10/11}\approx 91.51945222372814$
Ramified primes:   \(2\), \(7\), \(23\) 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:  \(644=2^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(363,·)$, $\chi_{644}(519,·)$, $\chi_{644}(265,·)$, $\chi_{644}(139,·)$, $\chi_{644}(13,·)$, $\chi_{644}(531,·)$, $\chi_{644}(533,·)$, $\chi_{644}(407,·)$, $\chi_{644}(27,·)$, $\chi_{644}(29,·)$, $\chi_{644}(545,·)$, $\chi_{644}(547,·)$, $\chi_{644}(167,·)$, $\chi_{644}(41,·)$, $\chi_{644}(561,·)$, $\chi_{644}(307,·)$, $\chi_{644}(393,·)$, $\chi_{644}(223,·)$, $\chi_{644}(279,·)$, $\chi_{644}(449,·)$, $\chi_{644}(323,·)$, $\chi_{644}(197,·)$, $\chi_{644}(71,·)$, $\chi_{644}(55,·)$, $\chi_{644}(461,·)$, $\chi_{644}(141,·)$, $\chi_{644}(209,·)$, $\chi_{644}(335,·)$, $\chi_{644}(211,·)$, $\chi_{644}(85,·)$, $\chi_{644}(601,·)$, $\chi_{644}(463,·)$, $\chi_{644}(349,·)$, $\chi_{644}(351,·)$, $\chi_{644}(225,·)$, $\chi_{644}(587,·)$, $\chi_{644}(489,·)$, $\chi_{644}(491,·)$, $\chi_{644}(239,·)$, $\chi_{644}(629,·)$, $\chi_{644}(169,·)$, $\chi_{644}(377,·)$, $\chi_{644}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{24}+\frac{1}{4}a^{22}-\frac{1}{2}a^{20}+\frac{1}{4}a^{18}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{25}+\frac{1}{8}a^{23}+\frac{1}{4}a^{21}+\frac{1}{8}a^{19}-\frac{1}{2}a^{15}+\frac{3}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{26}+\frac{1}{16}a^{24}+\frac{1}{8}a^{22}-\frac{7}{16}a^{20}-\frac{1}{2}a^{18}-\frac{1}{4}a^{16}-\frac{1}{2}a^{14}-\frac{5}{16}a^{12}+\frac{7}{16}a^{10}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{27}+\frac{1}{32}a^{25}+\frac{1}{16}a^{23}+\frac{9}{32}a^{21}+\frac{1}{4}a^{19}-\frac{1}{8}a^{17}-\frac{1}{4}a^{15}-\frac{5}{32}a^{13}-\frac{9}{32}a^{11}+\frac{15}{32}a^{9}-\frac{1}{16}a^{7}-\frac{15}{32}a^{5}-\frac{3}{8}a^{3}$, $\frac{1}{64}a^{28}+\frac{1}{64}a^{26}+\frac{1}{32}a^{24}-\frac{23}{64}a^{22}-\frac{3}{8}a^{20}+\frac{7}{16}a^{18}-\frac{1}{8}a^{16}-\frac{5}{64}a^{14}-\frac{9}{64}a^{12}+\frac{15}{64}a^{10}-\frac{1}{32}a^{8}-\frac{15}{64}a^{6}+\frac{5}{16}a^{4}$, $\frac{1}{128}a^{29}+\frac{1}{128}a^{27}+\frac{1}{64}a^{25}-\frac{23}{128}a^{23}-\frac{3}{16}a^{21}-\frac{9}{32}a^{19}+\frac{7}{16}a^{17}+\frac{59}{128}a^{15}-\frac{9}{128}a^{13}-\frac{49}{128}a^{11}-\frac{1}{64}a^{9}-\frac{15}{128}a^{7}-\frac{11}{32}a^{5}$, $\frac{1}{256}a^{30}+\frac{1}{256}a^{28}+\frac{1}{128}a^{26}-\frac{23}{256}a^{24}+\frac{13}{32}a^{22}+\frac{23}{64}a^{20}+\frac{7}{32}a^{18}-\frac{69}{256}a^{16}-\frac{9}{256}a^{14}+\frac{79}{256}a^{12}-\frac{1}{128}a^{10}+\frac{113}{256}a^{8}+\frac{21}{64}a^{6}$, $\frac{1}{512}a^{31}+\frac{1}{512}a^{29}+\frac{1}{256}a^{27}-\frac{23}{512}a^{25}+\frac{13}{64}a^{23}-\frac{41}{128}a^{21}+\frac{7}{64}a^{19}+\frac{187}{512}a^{17}-\frac{9}{512}a^{15}-\frac{177}{512}a^{13}-\frac{1}{256}a^{11}+\frac{113}{512}a^{9}+\frac{21}{128}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{32}+\frac{1}{1024}a^{30}+\frac{1}{512}a^{28}-\frac{23}{1024}a^{26}+\frac{13}{128}a^{24}-\frac{41}{256}a^{22}-\frac{57}{128}a^{20}+\frac{187}{1024}a^{18}+\frac{503}{1024}a^{16}-\frac{177}{1024}a^{14}-\frac{1}{512}a^{12}-\frac{399}{1024}a^{10}+\frac{21}{256}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{2048}a^{33}+\frac{1}{2048}a^{31}+\frac{1}{1024}a^{29}-\frac{23}{2048}a^{27}+\frac{13}{256}a^{25}-\frac{41}{512}a^{23}+\frac{71}{256}a^{21}+\frac{187}{2048}a^{19}+\frac{503}{2048}a^{17}+\frac{847}{2048}a^{15}-\frac{1}{1024}a^{13}+\frac{625}{2048}a^{11}+\frac{21}{512}a^{9}-\frac{1}{4}a^{7}-\frac{3}{8}a^{5}+\frac{3}{8}a^{3}$, $\frac{1}{4096}a^{34}+\frac{1}{4096}a^{32}+\frac{1}{2048}a^{30}-\frac{23}{4096}a^{28}+\frac{13}{512}a^{26}-\frac{41}{1024}a^{24}-\frac{185}{512}a^{22}+\frac{187}{4096}a^{20}+\frac{503}{4096}a^{18}-\frac{1201}{4096}a^{16}-\frac{1}{2048}a^{14}-\frac{1423}{4096}a^{12}+\frac{21}{1024}a^{10}-\frac{1}{8}a^{8}-\frac{3}{16}a^{6}+\frac{3}{16}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8192}a^{35}+\frac{1}{8192}a^{33}+\frac{1}{4096}a^{31}-\frac{23}{8192}a^{29}+\frac{13}{1024}a^{27}-\frac{41}{2048}a^{25}-\frac{185}{1024}a^{23}-\frac{3909}{8192}a^{21}-\frac{3593}{8192}a^{19}+\frac{2895}{8192}a^{17}-\frac{1}{4096}a^{15}+\frac{2673}{8192}a^{13}+\frac{21}{2048}a^{11}-\frac{1}{16}a^{9}+\frac{13}{32}a^{7}+\frac{3}{32}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{16384}a^{36}+\frac{1}{16384}a^{34}+\frac{1}{8192}a^{32}-\frac{23}{16384}a^{30}+\frac{13}{2048}a^{28}-\frac{41}{4096}a^{26}-\frac{185}{2048}a^{24}+\frac{4283}{16384}a^{22}+\frac{4599}{16384}a^{20}-\frac{5297}{16384}a^{18}+\frac{4095}{8192}a^{16}+\frac{2673}{16384}a^{14}-\frac{2027}{4096}a^{12}+\frac{15}{32}a^{10}+\frac{13}{64}a^{8}+\frac{3}{64}a^{6}+\frac{3}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32768}a^{37}+\frac{1}{32768}a^{35}+\frac{1}{16384}a^{33}-\frac{23}{32768}a^{31}+\frac{13}{4096}a^{29}-\frac{41}{8192}a^{27}-\frac{185}{4096}a^{25}+\frac{4283}{32768}a^{23}+\frac{4599}{32768}a^{21}-\frac{5297}{32768}a^{19}+\frac{4095}{16384}a^{17}+\frac{2673}{32768}a^{15}-\frac{2027}{8192}a^{13}-\frac{17}{64}a^{11}-\frac{51}{128}a^{9}+\frac{3}{128}a^{7}+\frac{3}{16}a^{5}+\frac{1}{4}a^{3}$, $\frac{1}{65536}a^{38}+\frac{1}{65536}a^{36}+\frac{1}{32768}a^{34}-\frac{23}{65536}a^{32}+\frac{13}{8192}a^{30}-\frac{41}{16384}a^{28}-\frac{185}{8192}a^{26}+\frac{4283}{65536}a^{24}-\frac{28169}{65536}a^{22}-\frac{5297}{65536}a^{20}+\frac{4095}{32768}a^{18}+\frac{2673}{65536}a^{16}-\frac{2027}{16384}a^{14}+\frac{47}{128}a^{12}+\frac{77}{256}a^{10}+\frac{3}{256}a^{8}-\frac{13}{32}a^{6}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{131072}a^{39}+\frac{1}{131072}a^{37}+\frac{1}{65536}a^{35}-\frac{23}{131072}a^{33}+\frac{13}{16384}a^{31}-\frac{41}{32768}a^{29}-\frac{185}{16384}a^{27}+\frac{4283}{131072}a^{25}-\frac{28169}{131072}a^{23}-\frac{5297}{131072}a^{21}-\frac{28673}{65536}a^{19}-\frac{62863}{131072}a^{17}+\frac{14357}{32768}a^{15}-\frac{81}{256}a^{13}-\frac{179}{512}a^{11}-\frac{253}{512}a^{9}-\frac{13}{64}a^{7}-\frac{7}{16}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{157024256}a^{40}+\frac{485}{157024256}a^{38}-\frac{1173}{78512128}a^{36}-\frac{16031}{157024256}a^{34}+\frac{2915}{39256064}a^{32}+\frac{31611}{39256064}a^{30}-\frac{91251}{19628032}a^{28}+\frac{1254619}{157024256}a^{26}+\frac{8769795}{157024256}a^{24}-\frac{56729701}{157024256}a^{22}-\frac{32836459}{78512128}a^{20}-\frac{21168135}{157024256}a^{18}-\frac{4634197}{19628032}a^{16}+\frac{575677}{2453504}a^{14}+\frac{536641}{1226752}a^{12}+\frac{44517}{153344}a^{10}+\frac{9443}{19168}a^{8}+\frac{679}{4792}a^{6}+\frac{1005}{9584}a^{4}-\frac{473}{2396}a^{2}-\frac{61}{599}$, $\frac{1}{314048512}a^{41}+\frac{485}{314048512}a^{39}-\frac{1173}{157024256}a^{37}-\frac{16031}{314048512}a^{35}+\frac{2915}{78512128}a^{33}+\frac{31611}{78512128}a^{31}-\frac{91251}{39256064}a^{29}+\frac{1254619}{314048512}a^{27}+\frac{8769795}{314048512}a^{25}-\frac{56729701}{314048512}a^{23}+\frac{45675669}{157024256}a^{21}-\frac{21168135}{314048512}a^{19}+\frac{14993835}{39256064}a^{17}+\frac{575677}{4907008}a^{15}-\frac{690111}{2453504}a^{13}-\frac{108827}{306688}a^{11}+\frac{9443}{38336}a^{9}-\frac{4113}{9584}a^{7}+\frac{1005}{19168}a^{5}+\frac{1923}{4792}a^{3}-\frac{61}{1198}a$, $\frac{1}{51\!\cdots\!04}a^{42}+\frac{85\!\cdots\!29}{51\!\cdots\!04}a^{40}+\frac{29\!\cdots\!31}{25\!\cdots\!52}a^{38}+\frac{14\!\cdots\!05}{51\!\cdots\!04}a^{36}+\frac{32\!\cdots\!29}{12\!\cdots\!76}a^{34}-\frac{26\!\cdots\!45}{12\!\cdots\!76}a^{32}+\frac{60\!\cdots\!05}{64\!\cdots\!88}a^{30}+\frac{36\!\cdots\!95}{51\!\cdots\!04}a^{28}+\frac{88\!\cdots\!39}{51\!\cdots\!04}a^{26}+\frac{48\!\cdots\!39}{51\!\cdots\!04}a^{24}+\frac{98\!\cdots\!37}{25\!\cdots\!52}a^{22}-\frac{16\!\cdots\!27}{51\!\cdots\!04}a^{20}-\frac{66\!\cdots\!37}{32\!\cdots\!44}a^{18}-\frac{34\!\cdots\!45}{80\!\cdots\!36}a^{16}+\frac{31\!\cdots\!73}{80\!\cdots\!36}a^{14}-\frac{10\!\cdots\!05}{10\!\cdots\!92}a^{12}-\frac{29\!\cdots\!05}{25\!\cdots\!48}a^{10}-\frac{32\!\cdots\!85}{12\!\cdots\!24}a^{8}+\frac{78\!\cdots\!71}{31\!\cdots\!56}a^{6}+\frac{17\!\cdots\!61}{39\!\cdots\!32}a^{4}+\frac{25\!\cdots\!67}{19\!\cdots\!16}a^{2}-\frac{15\!\cdots\!21}{49\!\cdots\!29}$, $\frac{1}{10\!\cdots\!08}a^{43}+\frac{85\!\cdots\!29}{10\!\cdots\!08}a^{41}+\frac{29\!\cdots\!31}{51\!\cdots\!04}a^{39}+\frac{14\!\cdots\!05}{10\!\cdots\!08}a^{37}+\frac{32\!\cdots\!29}{25\!\cdots\!52}a^{35}-\frac{26\!\cdots\!45}{25\!\cdots\!52}a^{33}+\frac{60\!\cdots\!05}{12\!\cdots\!76}a^{31}+\frac{36\!\cdots\!95}{10\!\cdots\!08}a^{29}+\frac{88\!\cdots\!39}{10\!\cdots\!08}a^{27}+\frac{48\!\cdots\!39}{10\!\cdots\!08}a^{25}+\frac{98\!\cdots\!37}{51\!\cdots\!04}a^{23}-\frac{16\!\cdots\!27}{10\!\cdots\!08}a^{21}+\frac{25\!\cdots\!07}{64\!\cdots\!88}a^{19}-\frac{34\!\cdots\!45}{16\!\cdots\!72}a^{17}-\frac{49\!\cdots\!63}{16\!\cdots\!72}a^{15}-\frac{10\!\cdots\!05}{20\!\cdots\!84}a^{13}-\frac{29\!\cdots\!05}{50\!\cdots\!96}a^{11}+\frac{93\!\cdots\!39}{25\!\cdots\!48}a^{9}-\frac{23\!\cdots\!85}{63\!\cdots\!12}a^{7}+\frac{17\!\cdots\!61}{78\!\cdots\!64}a^{5}-\frac{17\!\cdots\!49}{39\!\cdots\!32}a^{3}-\frac{15\!\cdots\!21}{98\!\cdots\!58}a$ 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{12510636459605472541565569626671330282473}{150992405799960827353876616408260423340252987392} a^{43} + \frac{788155204883906868031788120452544671121787}{150992405799960827353876616408260423340252987392} a^{41} - \frac{12222412535911517858439487774581238522975351}{75496202899980413676938308204130211670126493696} a^{39} + \frac{489873602995256836274423571564526256474629943}{150992405799960827353876616408260423340252987392} a^{37} - \frac{1764268916566608477463675942263307363705141737}{37748101449990206838469154102065105835063246848} a^{35} + \frac{19299108616983551654477233489468805331906499097}{37748101449990206838469154102065105835063246848} a^{33} - \frac{82722860010164010797077671965520911777142553537}{18874050724995103419234577051032552917531623424} a^{31} + \frac{4530681285881169048042403143196267234741261292461}{150992405799960827353876616408260423340252987392} a^{29} - \frac{25040951085494340503193687459240778980623332353731}{150992405799960827353876616408260423340252987392} a^{27} + \frac{112310131461680477894701525764055678298568062198805}{150992405799960827353876616408260423340252987392} a^{25} - \frac{204519381185656251262172760208092959650243567058537}{75496202899980413676938308204130211670126493696} a^{23} + \frac{1205778192527570118676095653861767352138544479378399}{150992405799960827353876616408260423340252987392} a^{21} - \frac{178535194968578488033671148578361950303629091182629}{9437025362497551709617288525516276458765811712} a^{19} + \frac{336119866696916397332529375363045901014608593036917}{9437025362497551709617288525516276458765811712} a^{17} - \frac{7732566688932177300653210328933389763846098481381}{147453521289024245462770133211191819668215808} a^{15} + \frac{8708574449713093653424698475988502986352103631101}{147453521289024245462770133211191819668215808} a^{13} - \frac{227441830194098487198347307634251285861508385891}{4607922540282007670711566662849744364631744} a^{11} + \frac{1080689946012840053506408863981892699816057045763}{36863380322256061365692533302797954917053952} a^{9} - \frac{1674625443307585495222322843759386236156697387}{143997579383812739709736458214054511394742} a^{7} + \frac{6352936899219365866975289920305886325414067353}{2303961270141003835355783331424872182315872} a^{5} - \frac{47692887008831282151574289740134176165856751}{143997579383812739709736458214054511394742} a^{3} + \frac{958711397086014320239456325257015048634325}{143997579383812739709736458214054511394742} a \)  (order $4$) 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 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 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 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 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 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 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 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*x^2 + 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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.0.3393400820453274956705986794666660343.1, 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 $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/37.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ $22^{2}$

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
\(2\) Copy content Toggle raw display Deg $22$$2$$11$$22$
Deg $22$$2$$11$$22$
\(7\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) 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}$