↑

Properties

Label 44.0.116...625.3
Degree $44$
Signature $[0, 22]$
Discriminant $1.166\times 10^{83}$
Root discriminant \(77.25\)
Ramified primes $3,5,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)
 
gp: K = bnfinit(y^44 - y^43 - 3*y^42 + 7*y^41 + 5*y^40 - 33*y^39 + 13*y^38 + 119*y^37 - 171*y^36 - 305*y^35 + 989*y^34 + 231*y^33 - 4187*y^32 + 3263*y^31 + 13485*y^30 - 26537*y^29 - 27403*y^28 + 133551*y^27 - 23939*y^26 - 510265*y^25 + 606021*y^24 + 1435039*y^23 - 3859123*y^22 + 5740156*y^21 + 9696336*y^20 - 32656960*y^19 - 6128384*y^18 + 136756224*y^17 - 112242688*y^16 - 434782208*y^15 + 883752960*y^14 + 855375872*y^13 - 4390387712*y^12 + 968884224*y^11 + 16592666624*y^10 - 20468203520*y^9 - 45902462976*y^8 + 127775277056*y^7 + 55834574848*y^6 - 566935683072*y^5 + 343597383680*y^4 + 1924145348608*y^3 - 3298534883328*y^2 - 4398046511104*y + 17592186044416, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)
 

\( x^{44} - x^{43} - 3 x^{42} + 7 x^{41} + 5 x^{40} - 33 x^{39} + 13 x^{38} + 119 x^{37} + \cdots + 17592186044416 \) 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:   \(116\!\cdots\!625\) \(\medspace = 3^{22}\cdot 5^{22}\cdot 23^{42}\) 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:  \(77.25\)
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:  $3^{1/2}5^{1/2}23^{21/22}\approx 77.24613267922518$
Ramified primes:   \(3\), \(5\), \(23\) 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:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(134,·)$, $\chi_{345}(136,·)$, $\chi_{345}(269,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(284,·)$, $\chi_{345}(29,·)$, $\chi_{345}(286,·)$, $\chi_{345}(31,·)$, $\chi_{345}(164,·)$, $\chi_{345}(166,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(179,·)$, $\chi_{345}(181,·)$, $\chi_{345}(314,·)$, $\chi_{345}(59,·)$, $\chi_{345}(316,·)$, $\chi_{345}(61,·)$, $\chi_{345}(194,·)$, $\chi_{345}(196,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(76,·)$, $\chi_{345}(209,·)$, $\chi_{345}(211,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(91,·)$, $\chi_{345}(224,·)$, $\chi_{345}(226,·)$, $\chi_{345}(104,·)$, $\chi_{345}(106,·)$, $\chi_{345}(239,·)$, $\chi_{345}(241,·)$, $\chi_{345}(119,·)$, $\chi_{345}(121,·)$, $\chi_{345}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{15436492}a^{23}+\frac{1}{4}a^{22}-\frac{1}{4}a^{21}+\frac{1}{4}a^{20}-\frac{1}{4}a^{19}+\frac{1}{4}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1435039}{3859123}$, $\frac{1}{61745968}a^{24}-\frac{1}{61745968}a^{23}-\frac{1}{16}a^{22}-\frac{3}{16}a^{21}+\frac{7}{16}a^{20}+\frac{5}{16}a^{19}-\frac{1}{16}a^{18}-\frac{3}{16}a^{17}+\frac{7}{16}a^{16}+\frac{5}{16}a^{15}-\frac{1}{16}a^{14}-\frac{3}{16}a^{13}+\frac{7}{16}a^{12}+\frac{5}{16}a^{11}-\frac{1}{16}a^{10}-\frac{3}{16}a^{9}+\frac{7}{16}a^{8}+\frac{5}{16}a^{7}-\frac{1}{16}a^{6}-\frac{3}{16}a^{5}+\frac{7}{16}a^{4}+\frac{5}{16}a^{3}-\frac{1}{16}a^{2}+\frac{1435039}{15436492}a+\frac{606021}{3859123}$, $\frac{1}{246983872}a^{25}-\frac{1}{246983872}a^{24}-\frac{3}{246983872}a^{23}+\frac{29}{64}a^{22}-\frac{25}{64}a^{21}-\frac{27}{64}a^{20}-\frac{1}{64}a^{19}-\frac{19}{64}a^{18}+\frac{23}{64}a^{17}-\frac{11}{64}a^{16}-\frac{17}{64}a^{15}-\frac{3}{64}a^{14}+\frac{7}{64}a^{13}+\frac{5}{64}a^{12}+\frac{31}{64}a^{11}+\frac{13}{64}a^{10}-\frac{9}{64}a^{9}+\frac{21}{64}a^{8}+\frac{15}{64}a^{7}+\frac{29}{64}a^{6}-\frac{25}{64}a^{5}-\frac{27}{64}a^{4}-\frac{1}{64}a^{3}+\frac{1435039}{61745968}a^{2}+\frac{606021}{15436492}a-\frac{510265}{3859123}$, $\frac{1}{987935488}a^{26}-\frac{1}{987935488}a^{25}-\frac{3}{987935488}a^{24}+\frac{7}{987935488}a^{23}+\frac{103}{256}a^{22}+\frac{37}{256}a^{21}+\frac{63}{256}a^{20}+\frac{45}{256}a^{19}-\frac{41}{256}a^{18}+\frac{117}{256}a^{17}+\frac{47}{256}a^{16}-\frac{3}{256}a^{15}+\frac{71}{256}a^{14}-\frac{59}{256}a^{13}+\frac{31}{256}a^{12}-\frac{51}{256}a^{11}-\frac{73}{256}a^{10}+\frac{21}{256}a^{9}+\frac{15}{256}a^{8}-\frac{99}{256}a^{7}+\frac{39}{256}a^{6}+\frac{101}{256}a^{5}-\frac{1}{256}a^{4}+\frac{1435039}{246983872}a^{3}+\frac{606021}{61745968}a^{2}-\frac{510265}{15436492}a-\frac{23939}{3859123}$, $\frac{1}{3951741952}a^{27}-\frac{1}{3951741952}a^{26}-\frac{3}{3951741952}a^{25}+\frac{7}{3951741952}a^{24}+\frac{5}{3951741952}a^{23}+\frac{37}{1024}a^{22}-\frac{449}{1024}a^{21}+\frac{301}{1024}a^{20}+\frac{471}{1024}a^{19}+\frac{373}{1024}a^{18}-\frac{209}{1024}a^{17}-\frac{259}{1024}a^{16}+\frac{71}{1024}a^{15}-\frac{59}{1024}a^{14}-\frac{225}{1024}a^{13}+\frac{461}{1024}a^{12}+\frac{439}{1024}a^{11}-\frac{235}{1024}a^{10}-\frac{497}{1024}a^{9}+\frac{413}{1024}a^{8}-\frac{473}{1024}a^{7}-\frac{155}{1024}a^{6}-\frac{1}{1024}a^{5}+\frac{1435039}{987935488}a^{4}+\frac{606021}{246983872}a^{3}-\frac{510265}{61745968}a^{2}-\frac{23939}{15436492}a+\frac{133551}{3859123}$, $\frac{1}{15806967808}a^{28}-\frac{1}{15806967808}a^{27}-\frac{3}{15806967808}a^{26}+\frac{7}{15806967808}a^{25}+\frac{5}{15806967808}a^{24}-\frac{33}{15806967808}a^{23}+\frac{575}{4096}a^{22}-\frac{723}{4096}a^{21}-\frac{1577}{4096}a^{20}+\frac{373}{4096}a^{19}+\frac{1839}{4096}a^{18}+\frac{765}{4096}a^{17}+\frac{71}{4096}a^{16}+\frac{965}{4096}a^{15}-\frac{1249}{4096}a^{14}+\frac{1485}{4096}a^{13}-\frac{585}{4096}a^{12}-\frac{1259}{4096}a^{11}-\frac{497}{4096}a^{10}+\frac{1437}{4096}a^{9}+\frac{551}{4096}a^{8}+\frac{1893}{4096}a^{7}-\frac{1}{4096}a^{6}+\frac{1435039}{3951741952}a^{5}+\frac{606021}{987935488}a^{4}-\frac{510265}{246983872}a^{3}-\frac{23939}{61745968}a^{2}+\frac{133551}{15436492}a-\frac{27403}{3859123}$, $\frac{1}{63227871232}a^{29}-\frac{1}{63227871232}a^{28}-\frac{3}{63227871232}a^{27}+\frac{7}{63227871232}a^{26}+\frac{5}{63227871232}a^{25}-\frac{33}{63227871232}a^{24}+\frac{13}{63227871232}a^{23}-\frac{723}{16384}a^{22}-\frac{1577}{16384}a^{21}+\frac{4469}{16384}a^{20}+\frac{1839}{16384}a^{19}-\frac{3331}{16384}a^{18}-\frac{4025}{16384}a^{17}+\frac{965}{16384}a^{16}-\frac{1249}{16384}a^{15}-\frac{2611}{16384}a^{14}+\frac{7607}{16384}a^{13}+\frac{2837}{16384}a^{12}-\frac{497}{16384}a^{11}+\frac{5533}{16384}a^{10}-\frac{3545}{16384}a^{9}-\frac{2203}{16384}a^{8}-\frac{1}{16384}a^{7}+\frac{1435039}{15806967808}a^{6}+\frac{606021}{3951741952}a^{5}-\frac{510265}{987935488}a^{4}-\frac{23939}{246983872}a^{3}+\frac{133551}{61745968}a^{2}-\frac{27403}{15436492}a-\frac{26537}{3859123}$, $\frac{1}{252911484928}a^{30}-\frac{1}{252911484928}a^{29}-\frac{3}{252911484928}a^{28}+\frac{7}{252911484928}a^{27}+\frac{5}{252911484928}a^{26}-\frac{33}{252911484928}a^{25}+\frac{13}{252911484928}a^{24}+\frac{119}{252911484928}a^{23}+\frac{31191}{65536}a^{22}-\frac{28299}{65536}a^{21}-\frac{30929}{65536}a^{20}+\frac{13053}{65536}a^{19}-\frac{20409}{65536}a^{18}-\frac{31803}{65536}a^{17}-\frac{17633}{65536}a^{16}+\frac{13773}{65536}a^{15}-\frac{8777}{65536}a^{14}+\frac{19221}{65536}a^{13}+\frac{15887}{65536}a^{12}-\frac{27235}{65536}a^{11}+\frac{29223}{65536}a^{10}+\frac{14181}{65536}a^{9}-\frac{1}{65536}a^{8}+\frac{1435039}{63227871232}a^{7}+\frac{606021}{15806967808}a^{6}-\frac{510265}{3951741952}a^{5}-\frac{23939}{987935488}a^{4}+\frac{133551}{246983872}a^{3}-\frac{27403}{61745968}a^{2}-\frac{26537}{15436492}a+\frac{13485}{3859123}$, $\frac{1}{1011645939712}a^{31}-\frac{1}{1011645939712}a^{30}-\frac{3}{1011645939712}a^{29}+\frac{7}{1011645939712}a^{28}+\frac{5}{1011645939712}a^{27}-\frac{33}{1011645939712}a^{26}+\frac{13}{1011645939712}a^{25}+\frac{119}{1011645939712}a^{24}-\frac{171}{1011645939712}a^{23}+\frac{102773}{262144}a^{22}+\frac{34607}{262144}a^{21}+\frac{78589}{262144}a^{20}+\frac{45127}{262144}a^{19}-\frac{97339}{262144}a^{18}-\frac{83169}{262144}a^{17}-\frac{51763}{262144}a^{16}+\frac{122295}{262144}a^{15}+\frac{84757}{262144}a^{14}-\frac{49649}{262144}a^{13}-\frac{27235}{262144}a^{12}-\frac{36313}{262144}a^{11}-\frac{116891}{262144}a^{10}-\frac{1}{262144}a^{9}+\frac{1435039}{252911484928}a^{8}+\frac{606021}{63227871232}a^{7}-\frac{510265}{15806967808}a^{6}-\frac{23939}{3951741952}a^{5}+\frac{133551}{987935488}a^{4}-\frac{27403}{246983872}a^{3}-\frac{26537}{61745968}a^{2}+\frac{13485}{15436492}a+\frac{3263}{3859123}$, $\frac{1}{4046583758848}a^{32}-\frac{1}{4046583758848}a^{31}-\frac{3}{4046583758848}a^{30}+\frac{7}{4046583758848}a^{29}+\frac{5}{4046583758848}a^{28}-\frac{33}{4046583758848}a^{27}+\frac{13}{4046583758848}a^{26}+\frac{119}{4046583758848}a^{25}-\frac{171}{4046583758848}a^{24}-\frac{305}{4046583758848}a^{23}-\frac{227537}{1048576}a^{22}-\frac{183555}{1048576}a^{21}+\frac{45127}{1048576}a^{20}-\frac{359483}{1048576}a^{19}+\frac{178975}{1048576}a^{18}+\frac{210381}{1048576}a^{17}+\frac{122295}{1048576}a^{16}+\frac{84757}{1048576}a^{15}+\frac{474639}{1048576}a^{14}+\frac{234909}{1048576}a^{13}-\frac{36313}{1048576}a^{12}+\frac{145253}{1048576}a^{11}-\frac{1}{1048576}a^{10}+\frac{1435039}{1011645939712}a^{9}+\frac{606021}{252911484928}a^{8}-\frac{510265}{63227871232}a^{7}-\frac{23939}{15806967808}a^{6}+\frac{133551}{3951741952}a^{5}-\frac{27403}{987935488}a^{4}-\frac{26537}{246983872}a^{3}+\frac{13485}{61745968}a^{2}+\frac{3263}{15436492}a-\frac{4187}{3859123}$, $\frac{1}{16186335035392}a^{33}-\frac{1}{16186335035392}a^{32}-\frac{3}{16186335035392}a^{31}+\frac{7}{16186335035392}a^{30}+\frac{5}{16186335035392}a^{29}-\frac{33}{16186335035392}a^{28}+\frac{13}{16186335035392}a^{27}+\frac{119}{16186335035392}a^{26}-\frac{171}{16186335035392}a^{25}-\frac{305}{16186335035392}a^{24}+\frac{989}{16186335035392}a^{23}+\frac{1913597}{4194304}a^{22}-\frac{1003449}{4194304}a^{21}+\frac{1737669}{4194304}a^{20}-\frac{1918177}{4194304}a^{19}-\frac{838195}{4194304}a^{18}+\frac{122295}{4194304}a^{17}-\frac{963819}{4194304}a^{16}+\frac{474639}{4194304}a^{15}-\frac{813667}{4194304}a^{14}-\frac{1084889}{4194304}a^{13}+\frac{145253}{4194304}a^{12}-\frac{1}{4194304}a^{11}+\frac{1435039}{4046583758848}a^{10}+\frac{606021}{1011645939712}a^{9}-\frac{510265}{252911484928}a^{8}-\frac{23939}{63227871232}a^{7}+\frac{133551}{15806967808}a^{6}-\frac{27403}{3951741952}a^{5}-\frac{26537}{987935488}a^{4}+\frac{13485}{246983872}a^{3}+\frac{3263}{61745968}a^{2}-\frac{4187}{15436492}a+\frac{231}{3859123}$, $\frac{1}{64745340141568}a^{34}-\frac{1}{64745340141568}a^{33}-\frac{3}{64745340141568}a^{32}+\frac{7}{64745340141568}a^{31}+\frac{5}{64745340141568}a^{30}-\frac{33}{64745340141568}a^{29}+\frac{13}{64745340141568}a^{28}+\frac{119}{64745340141568}a^{27}-\frac{171}{64745340141568}a^{26}-\frac{305}{64745340141568}a^{25}+\frac{989}{64745340141568}a^{24}+\frac{231}{64745340141568}a^{23}-\frac{1003449}{16777216}a^{22}-\frac{6650939}{16777216}a^{21}-\frac{6112481}{16777216}a^{20}-\frac{838195}{16777216}a^{19}-\frac{8266313}{16777216}a^{18}-\frac{5158123}{16777216}a^{17}+\frac{4668943}{16777216}a^{16}-\frac{813667}{16777216}a^{15}-\frac{1084889}{16777216}a^{14}+\frac{4339557}{16777216}a^{13}-\frac{1}{16777216}a^{12}+\frac{1435039}{16186335035392}a^{11}+\frac{606021}{4046583758848}a^{10}-\frac{510265}{1011645939712}a^{9}-\frac{23939}{252911484928}a^{8}+\frac{133551}{63227871232}a^{7}-\frac{27403}{15806967808}a^{6}-\frac{26537}{3951741952}a^{5}+\frac{13485}{987935488}a^{4}+\frac{3263}{246983872}a^{3}-\frac{4187}{61745968}a^{2}+\frac{231}{15436492}a+\frac{989}{3859123}$, $\frac{1}{258981360566272}a^{35}-\frac{1}{258981360566272}a^{34}-\frac{3}{258981360566272}a^{33}+\frac{7}{258981360566272}a^{32}+\frac{5}{258981360566272}a^{31}-\frac{33}{258981360566272}a^{30}+\frac{13}{258981360566272}a^{29}+\frac{119}{258981360566272}a^{28}-\frac{171}{258981360566272}a^{27}-\frac{305}{258981360566272}a^{26}+\frac{989}{258981360566272}a^{25}+\frac{231}{258981360566272}a^{24}-\frac{4187}{258981360566272}a^{23}-\frac{6650939}{67108864}a^{22}+\frac{10664735}{67108864}a^{21}+\frac{15939021}{67108864}a^{20}+\frac{8510903}{67108864}a^{19}-\frac{5158123}{67108864}a^{18}-\frac{28885489}{67108864}a^{17}-\frac{17590883}{67108864}a^{16}-\frac{1084889}{67108864}a^{15}+\frac{4339557}{67108864}a^{14}-\frac{1}{67108864}a^{13}+\frac{1435039}{64745340141568}a^{12}+\frac{606021}{16186335035392}a^{11}-\frac{510265}{4046583758848}a^{10}-\frac{23939}{1011645939712}a^{9}+\frac{133551}{252911484928}a^{8}-\frac{27403}{63227871232}a^{7}-\frac{26537}{15806967808}a^{6}+\frac{13485}{3951741952}a^{5}+\frac{3263}{987935488}a^{4}-\frac{4187}{246983872}a^{3}+\frac{231}{61745968}a^{2}+\frac{989}{15436492}a-\frac{305}{3859123}$, $\frac{1}{10\!\cdots\!88}a^{36}-\frac{1}{10\!\cdots\!88}a^{35}-\frac{3}{10\!\cdots\!88}a^{34}+\frac{7}{10\!\cdots\!88}a^{33}+\frac{5}{10\!\cdots\!88}a^{32}-\frac{33}{10\!\cdots\!88}a^{31}+\frac{13}{10\!\cdots\!88}a^{30}+\frac{119}{10\!\cdots\!88}a^{29}-\frac{171}{10\!\cdots\!88}a^{28}-\frac{305}{10\!\cdots\!88}a^{27}+\frac{989}{10\!\cdots\!88}a^{26}+\frac{231}{10\!\cdots\!88}a^{25}-\frac{4187}{10\!\cdots\!88}a^{24}+\frac{3263}{10\!\cdots\!88}a^{23}+\frac{10664735}{268435456}a^{22}+\frac{15939021}{268435456}a^{21}-\frac{58597961}{268435456}a^{20}-\frac{5158123}{268435456}a^{19}-\frac{28885489}{268435456}a^{18}+\frac{49517981}{268435456}a^{17}+\frac{66023975}{268435456}a^{16}+\frac{4339557}{268435456}a^{15}-\frac{1}{268435456}a^{14}+\frac{1435039}{258981360566272}a^{13}+\frac{606021}{64745340141568}a^{12}-\frac{510265}{16186335035392}a^{11}-\frac{23939}{4046583758848}a^{10}+\frac{133551}{1011645939712}a^{9}-\frac{27403}{252911484928}a^{8}-\frac{26537}{63227871232}a^{7}+\frac{13485}{15806967808}a^{6}+\frac{3263}{3951741952}a^{5}-\frac{4187}{987935488}a^{4}+\frac{231}{246983872}a^{3}+\frac{989}{61745968}a^{2}-\frac{305}{15436492}a-\frac{171}{3859123}$, $\frac{1}{41\!\cdots\!52}a^{37}-\frac{1}{41\!\cdots\!52}a^{36}-\frac{3}{41\!\cdots\!52}a^{35}+\frac{7}{41\!\cdots\!52}a^{34}+\frac{5}{41\!\cdots\!52}a^{33}-\frac{33}{41\!\cdots\!52}a^{32}+\frac{13}{41\!\cdots\!52}a^{31}+\frac{119}{41\!\cdots\!52}a^{30}-\frac{171}{41\!\cdots\!52}a^{29}-\frac{305}{41\!\cdots\!52}a^{28}+\frac{989}{41\!\cdots\!52}a^{27}+\frac{231}{41\!\cdots\!52}a^{26}-\frac{4187}{41\!\cdots\!52}a^{25}+\frac{3263}{41\!\cdots\!52}a^{24}+\frac{13485}{41\!\cdots\!52}a^{23}+\frac{284374477}{1073741824}a^{22}-\frac{327033417}{1073741824}a^{21}+\frac{263277333}{1073741824}a^{20}-\frac{28885489}{1073741824}a^{19}+\frac{49517981}{1073741824}a^{18}+\frac{66023975}{1073741824}a^{17}-\frac{264095899}{1073741824}a^{16}-\frac{1}{1073741824}a^{15}+\frac{1435039}{10\!\cdots\!88}a^{14}+\frac{606021}{258981360566272}a^{13}-\frac{510265}{64745340141568}a^{12}-\frac{23939}{16186335035392}a^{11}+\frac{133551}{4046583758848}a^{10}-\frac{27403}{1011645939712}a^{9}-\frac{26537}{252911484928}a^{8}+\frac{13485}{63227871232}a^{7}+\frac{3263}{15806967808}a^{6}-\frac{4187}{3951741952}a^{5}+\frac{231}{987935488}a^{4}+\frac{989}{246983872}a^{3}-\frac{305}{61745968}a^{2}-\frac{171}{15436492}a+\frac{119}{3859123}$, $\frac{1}{16\!\cdots\!08}a^{38}-\frac{1}{16\!\cdots\!08}a^{37}-\frac{3}{16\!\cdots\!08}a^{36}+\frac{7}{16\!\cdots\!08}a^{35}+\frac{5}{16\!\cdots\!08}a^{34}-\frac{33}{16\!\cdots\!08}a^{33}+\frac{13}{16\!\cdots\!08}a^{32}+\frac{119}{16\!\cdots\!08}a^{31}-\frac{171}{16\!\cdots\!08}a^{30}-\frac{305}{16\!\cdots\!08}a^{29}+\frac{989}{16\!\cdots\!08}a^{28}+\frac{231}{16\!\cdots\!08}a^{27}-\frac{4187}{16\!\cdots\!08}a^{26}+\frac{3263}{16\!\cdots\!08}a^{25}+\frac{13485}{16\!\cdots\!08}a^{24}-\frac{26537}{16\!\cdots\!08}a^{23}+\frac{746708407}{4294967296}a^{22}-\frac{1884206315}{4294967296}a^{21}-\frac{1102627313}{4294967296}a^{20}+\frac{49517981}{4294967296}a^{19}+\frac{66023975}{4294967296}a^{18}-\frac{264095899}{4294967296}a^{17}-\frac{1}{4294967296}a^{16}+\frac{1435039}{41\!\cdots\!52}a^{15}+\frac{606021}{10\!\cdots\!88}a^{14}-\frac{510265}{258981360566272}a^{13}-\frac{23939}{64745340141568}a^{12}+\frac{133551}{16186335035392}a^{11}-\frac{27403}{4046583758848}a^{10}-\frac{26537}{1011645939712}a^{9}+\frac{13485}{252911484928}a^{8}+\frac{3263}{63227871232}a^{7}-\frac{4187}{15806967808}a^{6}+\frac{231}{3951741952}a^{5}+\frac{989}{987935488}a^{4}-\frac{305}{246983872}a^{3}-\frac{171}{61745968}a^{2}+\frac{119}{15436492}a+\frac{13}{3859123}$, $\frac{1}{66\!\cdots\!32}a^{39}-\frac{1}{66\!\cdots\!32}a^{38}-\frac{3}{66\!\cdots\!32}a^{37}+\frac{7}{66\!\cdots\!32}a^{36}+\frac{5}{66\!\cdots\!32}a^{35}-\frac{33}{66\!\cdots\!32}a^{34}+\frac{13}{66\!\cdots\!32}a^{33}+\frac{119}{66\!\cdots\!32}a^{32}-\frac{171}{66\!\cdots\!32}a^{31}-\frac{305}{66\!\cdots\!32}a^{30}+\frac{989}{66\!\cdots\!32}a^{29}+\frac{231}{66\!\cdots\!32}a^{28}-\frac{4187}{66\!\cdots\!32}a^{27}+\frac{3263}{66\!\cdots\!32}a^{26}+\frac{13485}{66\!\cdots\!32}a^{25}-\frac{26537}{66\!\cdots\!32}a^{24}-\frac{27403}{66\!\cdots\!32}a^{23}-\frac{6179173611}{17179869184}a^{22}+\frac{3192339983}{17179869184}a^{21}+\frac{4344485277}{17179869184}a^{20}+\frac{66023975}{17179869184}a^{19}-\frac{264095899}{17179869184}a^{18}-\frac{1}{17179869184}a^{17}+\frac{1435039}{16\!\cdots\!08}a^{16}+\frac{606021}{41\!\cdots\!52}a^{15}-\frac{510265}{10\!\cdots\!88}a^{14}-\frac{23939}{258981360566272}a^{13}+\frac{133551}{64745340141568}a^{12}-\frac{27403}{16186335035392}a^{11}-\frac{26537}{4046583758848}a^{10}+\frac{13485}{1011645939712}a^{9}+\frac{3263}{252911484928}a^{8}-\frac{4187}{63227871232}a^{7}+\frac{231}{15806967808}a^{6}+\frac{989}{3951741952}a^{5}-\frac{305}{987935488}a^{4}-\frac{171}{246983872}a^{3}+\frac{119}{61745968}a^{2}+\frac{13}{15436492}a-\frac{33}{3859123}$, $\frac{1}{26\!\cdots\!28}a^{40}-\frac{1}{26\!\cdots\!28}a^{39}-\frac{3}{26\!\cdots\!28}a^{38}+\frac{7}{26\!\cdots\!28}a^{37}+\frac{5}{26\!\cdots\!28}a^{36}-\frac{33}{26\!\cdots\!28}a^{35}+\frac{13}{26\!\cdots\!28}a^{34}+\frac{119}{26\!\cdots\!28}a^{33}-\frac{171}{26\!\cdots\!28}a^{32}-\frac{305}{26\!\cdots\!28}a^{31}+\frac{989}{26\!\cdots\!28}a^{30}+\frac{231}{26\!\cdots\!28}a^{29}-\frac{4187}{26\!\cdots\!28}a^{28}+\frac{3263}{26\!\cdots\!28}a^{27}+\frac{13485}{26\!\cdots\!28}a^{26}-\frac{26537}{26\!\cdots\!28}a^{25}-\frac{27403}{26\!\cdots\!28}a^{24}+\frac{133551}{26\!\cdots\!28}a^{23}+\frac{20372209167}{68719476736}a^{22}+\frac{4344485277}{68719476736}a^{21}-\frac{17113845209}{68719476736}a^{20}-\frac{264095899}{68719476736}a^{19}-\frac{1}{68719476736}a^{18}+\frac{1435039}{66\!\cdots\!32}a^{17}+\frac{606021}{16\!\cdots\!08}a^{16}-\frac{510265}{41\!\cdots\!52}a^{15}-\frac{23939}{10\!\cdots\!88}a^{14}+\frac{133551}{258981360566272}a^{13}-\frac{27403}{64745340141568}a^{12}-\frac{26537}{16186335035392}a^{11}+\frac{13485}{4046583758848}a^{10}+\frac{3263}{1011645939712}a^{9}-\frac{4187}{252911484928}a^{8}+\frac{231}{63227871232}a^{7}+\frac{989}{15806967808}a^{6}-\frac{305}{3951741952}a^{5}-\frac{171}{987935488}a^{4}+\frac{119}{246983872}a^{3}+\frac{13}{61745968}a^{2}-\frac{33}{15436492}a+\frac{5}{3859123}$, $\frac{1}{10\!\cdots\!12}a^{41}-\frac{1}{10\!\cdots\!12}a^{40}-\frac{3}{10\!\cdots\!12}a^{39}+\frac{7}{10\!\cdots\!12}a^{38}+\frac{5}{10\!\cdots\!12}a^{37}-\frac{33}{10\!\cdots\!12}a^{36}+\frac{13}{10\!\cdots\!12}a^{35}+\frac{119}{10\!\cdots\!12}a^{34}-\frac{171}{10\!\cdots\!12}a^{33}-\frac{305}{10\!\cdots\!12}a^{32}+\frac{989}{10\!\cdots\!12}a^{31}+\frac{231}{10\!\cdots\!12}a^{30}-\frac{4187}{10\!\cdots\!12}a^{29}+\frac{3263}{10\!\cdots\!12}a^{28}+\frac{13485}{10\!\cdots\!12}a^{27}-\frac{26537}{10\!\cdots\!12}a^{26}-\frac{27403}{10\!\cdots\!12}a^{25}+\frac{133551}{10\!\cdots\!12}a^{24}-\frac{23939}{10\!\cdots\!12}a^{23}+\frac{4344485277}{274877906944}a^{22}-\frac{85833321945}{274877906944}a^{21}+\frac{68455380837}{274877906944}a^{20}-\frac{1}{274877906944}a^{19}+\frac{1435039}{26\!\cdots\!28}a^{18}+\frac{606021}{66\!\cdots\!32}a^{17}-\frac{510265}{16\!\cdots\!08}a^{16}-\frac{23939}{41\!\cdots\!52}a^{15}+\frac{133551}{10\!\cdots\!88}a^{14}-\frac{27403}{258981360566272}a^{13}-\frac{26537}{64745340141568}a^{12}+\frac{13485}{16186335035392}a^{11}+\frac{3263}{4046583758848}a^{10}-\frac{4187}{1011645939712}a^{9}+\frac{231}{252911484928}a^{8}+\frac{989}{63227871232}a^{7}-\frac{305}{15806967808}a^{6}-\frac{171}{3951741952}a^{5}+\frac{119}{987935488}a^{4}+\frac{13}{246983872}a^{3}-\frac{33}{61745968}a^{2}+\frac{5}{15436492}a+\frac{7}{3859123}$, $\frac{1}{42\!\cdots\!48}a^{42}-\frac{1}{42\!\cdots\!48}a^{41}-\frac{3}{42\!\cdots\!48}a^{40}+\frac{7}{42\!\cdots\!48}a^{39}+\frac{5}{42\!\cdots\!48}a^{38}-\frac{33}{42\!\cdots\!48}a^{37}+\frac{13}{42\!\cdots\!48}a^{36}+\frac{119}{42\!\cdots\!48}a^{35}-\frac{171}{42\!\cdots\!48}a^{34}-\frac{305}{42\!\cdots\!48}a^{33}+\frac{989}{42\!\cdots\!48}a^{32}+\frac{231}{42\!\cdots\!48}a^{31}-\frac{4187}{42\!\cdots\!48}a^{30}+\frac{3263}{42\!\cdots\!48}a^{29}+\frac{13485}{42\!\cdots\!48}a^{28}-\frac{26537}{42\!\cdots\!48}a^{27}-\frac{27403}{42\!\cdots\!48}a^{26}+\frac{133551}{42\!\cdots\!48}a^{25}-\frac{23939}{42\!\cdots\!48}a^{24}-\frac{510265}{42\!\cdots\!48}a^{23}-\frac{360711228889}{1099511627776}a^{22}+\frac{343333287781}{1099511627776}a^{21}-\frac{1}{1099511627776}a^{20}+\frac{1435039}{10\!\cdots\!12}a^{19}+\frac{606021}{26\!\cdots\!28}a^{18}-\frac{510265}{66\!\cdots\!32}a^{17}-\frac{23939}{16\!\cdots\!08}a^{16}+\frac{133551}{41\!\cdots\!52}a^{15}-\frac{27403}{10\!\cdots\!88}a^{14}-\frac{26537}{258981360566272}a^{13}+\frac{13485}{64745340141568}a^{12}+\frac{3263}{16186335035392}a^{11}-\frac{4187}{4046583758848}a^{10}+\frac{231}{1011645939712}a^{9}+\frac{989}{252911484928}a^{8}-\frac{305}{63227871232}a^{7}-\frac{171}{15806967808}a^{6}+\frac{119}{3951741952}a^{5}+\frac{13}{987935488}a^{4}-\frac{33}{246983872}a^{3}+\frac{5}{61745968}a^{2}+\frac{7}{15436492}a-\frac{3}{3859123}$, $\frac{1}{16\!\cdots\!92}a^{43}-\frac{1}{16\!\cdots\!92}a^{42}-\frac{3}{16\!\cdots\!92}a^{41}+\frac{7}{16\!\cdots\!92}a^{40}+\frac{5}{16\!\cdots\!92}a^{39}-\frac{33}{16\!\cdots\!92}a^{38}+\frac{13}{16\!\cdots\!92}a^{37}+\frac{119}{16\!\cdots\!92}a^{36}-\frac{171}{16\!\cdots\!92}a^{35}-\frac{305}{16\!\cdots\!92}a^{34}+\frac{989}{16\!\cdots\!92}a^{33}+\frac{231}{16\!\cdots\!92}a^{32}-\frac{4187}{16\!\cdots\!92}a^{31}+\frac{3263}{16\!\cdots\!92}a^{30}+\frac{13485}{16\!\cdots\!92}a^{29}-\frac{26537}{16\!\cdots\!92}a^{28}-\frac{27403}{16\!\cdots\!92}a^{27}+\frac{133551}{16\!\cdots\!92}a^{26}-\frac{23939}{16\!\cdots\!92}a^{25}-\frac{510265}{16\!\cdots\!92}a^{24}+\frac{606021}{16\!\cdots\!92}a^{23}+\frac{1442844915557}{4398046511104}a^{22}-\frac{1}{4398046511104}a^{21}+\frac{1435039}{42\!\cdots\!48}a^{20}+\frac{606021}{10\!\cdots\!12}a^{19}-\frac{510265}{26\!\cdots\!28}a^{18}-\frac{23939}{66\!\cdots\!32}a^{17}+\frac{133551}{16\!\cdots\!08}a^{16}-\frac{27403}{41\!\cdots\!52}a^{15}-\frac{26537}{10\!\cdots\!88}a^{14}+\frac{13485}{258981360566272}a^{13}+\frac{3263}{64745340141568}a^{12}-\frac{4187}{16186335035392}a^{11}+\frac{231}{4046583758848}a^{10}+\frac{989}{1011645939712}a^{9}-\frac{305}{252911484928}a^{8}-\frac{171}{63227871232}a^{7}+\frac{119}{15806967808}a^{6}+\frac{13}{3951741952}a^{5}-\frac{33}{987935488}a^{4}+\frac{5}{246983872}a^{3}+\frac{7}{61745968}a^{2}-\frac{3}{15436492}a-\frac{1}{3859123}$ 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:   \( -\frac{989}{16186335035392} a^{34} - \frac{6568471697}{16186335035392} a^{11} \)  (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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)
 
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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416, 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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416);
 
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 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416);
 
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

\(\Q(\sqrt{345}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.341419566026798986253349758444608447265625.1, \(\Q(\zeta_{23})\), 22.0.14844328957686912445797815584548193359375.1

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 ${\href{/padicField/2.11.0.1}{11} }^{4}$ R R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ $22^{2}$ $22^{2}$

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 3.22.11.1$x^{22} - 39366 x^{4} - 177147$$2$$11$$11$22T1$[\ ]_{2}^{11}$
3.22.11.1$x^{22} - 39366 x^{4} - 177147$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(5\) 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}$