Properties

Label 44.0.116...625.2
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)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)
 
gp: K = bnfinit(y^44 - y^43 - 23*y^42 + 24*y^41 + 298*y^40 - 322*y^39 - 2644*y^38 + 2966*y^37 + 17733*y^36 - 20699*y^35 - 93680*y^34 + 114379*y^33 + 400546*y^32 - 514925*y^31 - 1402310*y^30 + 1917235*y^29 + 4042594*y^28 - 5959829*y^27 - 9550726*y^26 + 15510555*y^25 + 18239299*y^24 - 33749854*y^23 - 27151731*y^22 + 60965664*y^21 + 28718628*y^20 - 92631926*y^19 - 11478724*y^18 + 118848820*y^17 - 33434940*y^16 - 1406311*y^15 - 22927136*y^14 - 576983889*y^13 + 635212253*y^12 + 643308528*y^11 - 1294716116*y^10 + 1724346364*y^9 - 424183664*y^8 - 3407721010*y^7 + 3830694322*y^6 + 209294181*y^5 - 4039808114*y^4 + 4041269764*y^3 - 1473288*y^2 - 4104644424*y + 4106118241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)
 

\( x^{44} - x^{43} - 23 x^{42} + 24 x^{41} + 298 x^{40} - 322 x^{39} - 2644 x^{38} + 2966 x^{37} + 17733 x^{36} - 20699 x^{35} - 93680 x^{34} + 114379 x^{33} + 400546 x^{32} + \cdots + 4106118241 \) 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$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(131,·)$, $\chi_{345}(134,·)$, $\chi_{345}(266,·)$, $\chi_{345}(109,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(146,·)$, $\chi_{345}(19,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(26,·)$, $\chi_{345}(31,·)$, $\chi_{345}(34,·)$, $\chi_{345}(41,·)$, $\chi_{345}(71,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(304,·)$, $\chi_{345}(311,·)$, $\chi_{345}(116,·)$, $\chi_{345}(314,·)$, $\chi_{345}(319,·)$, $\chi_{345}(194,·)$, $\chi_{345}(196,·)$, $\chi_{345}(326,·)$, $\chi_{345}(199,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(79,·)$, $\chi_{345}(211,·)$, $\chi_{345}(214,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(101,·)$, $\chi_{345}(224,·)$, $\chi_{345}(229,·)$, $\chi_{345}(236,·)$, $\chi_{345}(274,·)$, $\chi_{345}(244,·)$, $\chi_{345}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{28657}a^{24}+\frac{10946}{28657}a^{23}-\frac{13}{28657}a^{22}+\frac{5168}{28657}a^{21}+\frac{101}{28657}a^{20}-\frac{754}{28657}a^{19}-\frac{501}{28657}a^{18}-\frac{7930}{28657}a^{17}+\frac{1797}{28657}a^{16}+\frac{11857}{28657}a^{15}-\frac{4598}{28657}a^{14}-\frac{3796}{28657}a^{13}+\frac{8635}{28657}a^{12}+\frac{9498}{28657}a^{11}-\frac{11284}{28657}a^{10}+\frac{14025}{28657}a^{9}+\frac{10152}{28657}a^{8}+\frac{12302}{28657}a^{7}-\frac{5510}{28657}a^{6}-\frac{6477}{28657}a^{5}+\frac{1691}{28657}a^{4}+\frac{6477}{28657}a^{3}-\frac{193}{28657}a^{2}+\frac{11556}{28657}a+\frac{4}{28657}$, $\frac{1}{28657}a^{25}-\frac{12}{28657}a^{23}+\frac{4181}{28657}a^{22}+\frac{91}{28657}a^{21}+\frac{11323}{28657}a^{20}-\frac{433}{28657}a^{19}+\frac{2529}{28657}a^{18}+\frac{1524}{28657}a^{17}+\frac{597}{28657}a^{16}-\frac{3767}{28657}a^{15}+\frac{4220}{28657}a^{14}+\frac{7001}{28657}a^{13}+\frac{1574}{28657}a^{12}-\frac{8796}{28657}a^{11}-\frac{11638}{28657}a^{10}+\frac{8051}{28657}a^{9}-\frac{8301}{28657}a^{8}-\frac{3959}{28657}a^{7}+\frac{11655}{28657}a^{6}+\frac{1515}{28657}a^{5}+\frac{9213}{28657}a^{4}-\frac{17}{28657}a^{3}+\frac{3516}{28657}a^{2}+\frac{26}{28657}a+\frac{13530}{28657}$, $\frac{1}{28657}a^{26}-\frac{7752}{28657}a^{23}-\frac{65}{28657}a^{22}-\frac{12632}{28657}a^{21}+\frac{779}{28657}a^{20}-\frac{6519}{28657}a^{19}-\frac{4488}{28657}a^{18}-\frac{8592}{28657}a^{17}-\frac{10860}{28657}a^{16}+\frac{3219}{28657}a^{15}+\frac{9139}{28657}a^{14}+\frac{13336}{28657}a^{13}+\frac{8853}{28657}a^{12}-\frac{12290}{28657}a^{11}-\frac{12729}{28657}a^{10}-\frac{11943}{28657}a^{9}+\frac{3237}{28657}a^{8}-\frac{12663}{28657}a^{7}-\frac{7291}{28657}a^{6}-\frac{11197}{28657}a^{5}-\frac{8382}{28657}a^{4}-\frac{4731}{28657}a^{3}-\frac{2290}{28657}a^{2}+\frac{8917}{28657}a+\frac{48}{28657}$, $\frac{1}{28657}a^{27}-\frac{50}{28657}a^{23}+\frac{1220}{28657}a^{22}+\frac{629}{28657}a^{21}+\frac{2694}{28657}a^{20}-\frac{3468}{28657}a^{19}+\frac{5008}{28657}a^{18}+\frac{13702}{28657}a^{17}+\frac{6261}{28657}a^{16}-\frac{7053}{28657}a^{15}-\frac{9709}{28657}a^{14}+\frac{13000}{28657}a^{13}+\frac{12135}{28657}a^{12}-\frac{4066}{28657}a^{11}+\frac{4310}{28657}a^{10}+\frac{379}{28657}a^{9}-\frac{6481}{28657}a^{8}-\frac{12683}{28657}a^{7}+\frac{2870}{28657}a^{6}-\frac{11022}{28657}a^{5}+\frac{7652}{28657}a^{4}+\frac{350}{28657}a^{3}+\frac{2945}{28657}a^{2}+\frac{378}{28657}a+\frac{2351}{28657}$, $\frac{1}{28657}a^{28}+\frac{4037}{28657}a^{23}-\frac{21}{28657}a^{22}+\frac{3181}{28657}a^{21}+\frac{1582}{28657}a^{20}-\frac{4035}{28657}a^{19}-\frac{11348}{28657}a^{18}+\frac{10959}{28657}a^{17}-\frac{3174}{28657}a^{16}+\frac{10001}{28657}a^{15}+\frac{12356}{28657}a^{14}-\frac{5723}{28657}a^{13}-\frac{2171}{28657}a^{12}-\frac{7959}{28657}a^{11}+\frac{9319}{28657}a^{10}+\frac{7001}{28657}a^{9}+\frac{7748}{28657}a^{8}-\frac{12484}{28657}a^{7}+\frac{48}{28657}a^{6}-\frac{971}{28657}a^{5}-\frac{1071}{28657}a^{4}+\frac{11568}{28657}a^{3}-\frac{9272}{28657}a^{2}+\frac{7011}{28657}a+\frac{200}{28657}$, $\frac{1}{28657}a^{29}+\frac{71}{28657}a^{23}-\frac{1652}{28657}a^{22}+\frac{662}{28657}a^{21}-\frac{10574}{28657}a^{20}-\frac{5092}{28657}a^{19}-\frac{1151}{28657}a^{18}+\frac{367}{28657}a^{17}+\frac{5733}{28657}a^{16}+\frac{2837}{28657}a^{15}-\frac{13333}{28657}a^{14}-\frac{9214}{28657}a^{13}+\frac{8115}{28657}a^{12}+\frac{8959}{28657}a^{11}-\frac{4121}{28657}a^{10}-\frac{13602}{28657}a^{9}+\frac{12059}{28657}a^{8}-\frac{545}{28657}a^{7}+\frac{5067}{28657}a^{6}+\frac{11394}{28657}a^{5}+\frac{5367}{28657}a^{4}+\frac{6920}{28657}a^{3}+\frac{12413}{28657}a^{2}+\frac{2224}{28657}a+\frac{12509}{28657}$, $\frac{1}{28657}a^{30}-\frac{5079}{28657}a^{23}+\frac{1585}{28657}a^{22}-\frac{4961}{28657}a^{21}-\frac{12263}{28657}a^{20}-\frac{4931}{28657}a^{19}+\frac{7281}{28657}a^{18}-\frac{4377}{28657}a^{17}-\frac{10122}{28657}a^{16}+\frac{4530}{28657}a^{15}+\frac{2017}{28657}a^{14}-\frac{8939}{28657}a^{13}-\frac{2329}{28657}a^{12}+\frac{9289}{28657}a^{11}+\frac{13823}{28657}a^{10}-\frac{9378}{28657}a^{9}-\frac{4912}{28657}a^{8}-\frac{8665}{28657}a^{7}+\frac{1406}{28657}a^{6}+\frac{6722}{28657}a^{5}+\frac{1487}{28657}a^{4}+\frac{11058}{28657}a^{3}-\frac{12730}{28657}a^{2}-\frac{5571}{28657}a-\frac{284}{28657}$, $\frac{1}{28657}a^{31}+\frac{1739}{28657}a^{23}-\frac{13674}{28657}a^{22}-\frac{13803}{28657}a^{21}-\frac{7778}{28657}a^{20}-\frac{10904}{28657}a^{19}+\frac{1517}{28657}a^{18}+\frac{5150}{28657}a^{17}-\frac{10090}{28657}a^{16}-\frac{13294}{28657}a^{15}-\frac{6726}{28657}a^{14}+\frac{3948}{28657}a^{13}-\frac{7413}{28657}a^{12}-\frac{4223}{28657}a^{11}-\frac{6814}{28657}a^{10}-\frac{13239}{28657}a^{9}-\frac{600}{28657}a^{8}+\frac{11004}{28657}a^{7}-\frac{9336}{28657}a^{6}+\frac{3040}{28657}a^{5}+\frac{2547}{28657}a^{4}-\frac{14283}{28657}a^{3}-\frac{11480}{28657}a^{2}+\frac{3104}{28657}a-\frac{8341}{28657}$, $\frac{1}{28657}a^{32}+\frac{8137}{28657}a^{23}+\frac{8804}{28657}a^{22}+\frac{3368}{28657}a^{21}+\frac{14056}{28657}a^{20}-\frac{5499}{28657}a^{19}-\frac{11978}{28657}a^{18}-\frac{3837}{28657}a^{17}+\frac{13993}{28657}a^{16}+\frac{6991}{28657}a^{15}+\frac{4567}{28657}a^{14}+\frac{2721}{28657}a^{13}-\frac{4220}{28657}a^{12}+\frac{11253}{28657}a^{11}+\frac{8249}{28657}a^{10}-\frac{2968}{28657}a^{9}+\frac{9388}{28657}a^{8}+\frac{4265}{28657}a^{7}+\frac{13492}{28657}a^{6}+\frac{3849}{28657}a^{5}-\frac{3261}{28657}a^{4}-\frac{12782}{28657}a^{3}-\frac{5153}{28657}a^{2}+\frac{12989}{28657}a-\frac{6956}{28657}$, $\frac{1}{28657}a^{33}+\frac{7158}{28657}a^{23}-\frac{5479}{28657}a^{22}+\frac{1859}{28657}a^{21}+\frac{3717}{28657}a^{20}-\frac{9278}{28657}a^{19}+\frac{3506}{28657}a^{18}+\frac{4839}{28657}a^{17}-\frac{128}{28657}a^{16}+\frac{12277}{28657}a^{15}-\frac{9395}{28657}a^{14}-\frac{8414}{28657}a^{13}-\frac{13435}{28657}a^{12}+\frac{10952}{28657}a^{11}-\frac{2088}{28657}a^{10}+\frac{137}{28657}a^{9}-\frac{13085}{28657}a^{8}+\frac{11019}{28657}a^{7}-\frac{9486}{28657}a^{6}-\frac{135}{28657}a^{5}+\frac{11568}{28657}a^{4}-\frac{8279}{28657}a^{3}+\frac{7295}{28657}a^{2}+\frac{14146}{28657}a-\frac{3891}{28657}$, $\frac{1}{28657}a^{34}-\frac{8709}{28657}a^{23}+\frac{8942}{28657}a^{22}+\frac{7360}{28657}a^{21}+\frac{12846}{28657}a^{20}+\frac{13122}{28657}a^{19}+\frac{8872}{28657}a^{18}-\frac{6705}{28657}a^{17}-\frac{12313}{28657}a^{16}+\frac{233}{28657}a^{15}+\frac{5834}{28657}a^{14}-\frac{8503}{28657}a^{13}-\frac{13886}{28657}a^{12}+\frac{14289}{28657}a^{11}-\frac{13074}{28657}a^{10}+\frac{10093}{28657}a^{9}-\frac{11502}{28657}a^{8}-\frac{4241}{28657}a^{7}+\frac{8413}{28657}a^{6}+\frac{6908}{28657}a^{5}+\frac{9454}{28657}a^{4}+\frac{11955}{28657}a^{3}-\frac{8553}{28657}a^{2}+\frac{11020}{28657}a+\frac{25}{28657}$, $\frac{1}{28657}a^{35}-\frac{4183}{28657}a^{23}+\frac{8771}{28657}a^{22}+\frac{811}{28657}a^{21}+\frac{4364}{28657}a^{20}+\frac{4739}{28657}a^{19}-\frac{14050}{28657}a^{18}-\frac{11313}{28657}a^{17}+\frac{3584}{28657}a^{16}-\frac{11381}{28657}a^{15}+\frac{10001}{28657}a^{14}-\frac{3072}{28657}a^{13}-\frac{8121}{28657}a^{12}+\frac{906}{28657}a^{11}+\frac{2590}{28657}a^{10}-\frac{3911}{28657}a^{9}+\frac{2682}{28657}a^{8}-\frac{1992}{28657}a^{7}-\frac{7864}{28657}a^{6}-\frac{1763}{28657}a^{5}+\frac{9176}{28657}a^{4}+\frac{2664}{28657}a^{3}-\frac{7711}{28657}a^{2}-\frac{2155}{28657}a+\frac{6179}{28657}$, $\frac{1}{28657}a^{36}+\frac{2003}{28657}a^{23}+\frac{3746}{28657}a^{22}-\frac{13927}{28657}a^{21}-\frac{2633}{28657}a^{20}+\frac{12895}{28657}a^{19}+\frac{13622}{28657}a^{18}-\frac{11457}{28657}a^{17}-\frac{2664}{28657}a^{16}+\frac{2565}{28657}a^{15}-\frac{7659}{28657}a^{14}-\frac{10811}{28657}a^{13}+\frac{13291}{28657}a^{12}+\frac{14122}{28657}a^{11}-\frac{6804}{28657}a^{10}+\frac{8378}{28657}a^{9}-\frac{5850}{28657}a^{8}+\frac{12087}{28657}a^{7}-\frac{9865}{28657}a^{6}-\frac{3250}{28657}a^{5}-\frac{2162}{28657}a^{4}+\frac{4715}{28657}a^{3}-\frac{7078}{28657}a^{2}+\frac{568}{28657}a-\frac{11925}{28657}$, $\frac{1}{28657}a^{37}+\frac{1513}{28657}a^{23}+\frac{12112}{28657}a^{22}-\frac{8960}{28657}a^{21}+\frac{11191}{28657}a^{20}+\frac{5063}{28657}a^{19}-\frac{10949}{28657}a^{18}+\frac{5148}{28657}a^{17}+\frac{13956}{28657}a^{16}-\frac{577}{28657}a^{15}+\frac{86}{28657}a^{14}-\frac{6083}{28657}a^{13}-\frac{1612}{28657}a^{12}-\frac{3050}{28657}a^{11}-\frac{143}{28657}a^{10}-\frac{14065}{28657}a^{9}-\frac{4556}{28657}a^{8}-\frac{5751}{28657}a^{7}+\frac{335}{28657}a^{6}-\frac{10352}{28657}a^{5}-\frac{832}{28657}a^{4}+\frac{1112}{28657}a^{3}-\frac{14051}{28657}a^{2}-\frac{3737}{28657}a-\frac{8012}{28657}$, $\frac{1}{28657}a^{38}-\frac{14097}{28657}a^{23}+\frac{10709}{28657}a^{22}-\frac{13289}{28657}a^{21}-\frac{4465}{28657}a^{20}+\frac{12230}{28657}a^{19}-\frac{10578}{28657}a^{18}+\frac{4763}{28657}a^{17}+\frac{2977}{28657}a^{16}-\frac{273}{28657}a^{15}-\frac{12960}{28657}a^{14}+\frac{10336}{28657}a^{13}-\frac{213}{28657}a^{12}-\frac{13460}{28657}a^{11}+\frac{7712}{28657}a^{10}+\frac{10456}{28657}a^{9}-\frac{5575}{28657}a^{8}-\frac{14198}{28657}a^{7}-\frac{12909}{28657}a^{6}-\frac{1825}{28657}a^{5}-\frac{6898}{28657}a^{4}-\frac{13058}{28657}a^{3}+\frac{1702}{28657}a^{2}-\frac{11470}{28657}a-\frac{6052}{28657}$, $\frac{1}{28657}a^{39}-\frac{1474}{28657}a^{23}+\frac{4049}{28657}a^{22}+\frac{2737}{28657}a^{21}+\frac{3177}{28657}a^{20}-\frac{7969}{28657}a^{19}-\frac{8212}{28657}a^{18}+\frac{4724}{28657}a^{17}-\frac{752}{28657}a^{16}+\frac{7545}{28657}a^{15}-\frac{14193}{28657}a^{14}-\frac{9806}{28657}a^{13}+\frac{7856}{28657}a^{12}-\frac{13143}{28657}a^{11}-\frac{13742}{28657}a^{10}+\frac{207}{28657}a^{9}+\frac{14145}{28657}a^{8}+\frac{4878}{28657}a^{7}+\frac{12832}{28657}a^{6}-\frac{11965}{28657}a^{5}+\frac{11002}{28657}a^{4}+\frac{6769}{28657}a^{3}-\frac{9776}{28657}a^{2}+\frac{12492}{28657}a-\frac{926}{28657}$, $\frac{1}{3983323}a^{40}-\frac{43}{3983323}a^{39}-\frac{47}{3983323}a^{38}-\frac{64}{3983323}a^{37}+\frac{13}{3983323}a^{36}-\frac{3}{3983323}a^{35}-\frac{68}{3983323}a^{34}+\frac{5}{3983323}a^{33}+\frac{23}{3983323}a^{32}-\frac{16}{3983323}a^{31}+\frac{45}{3983323}a^{30}+\frac{11}{3983323}a^{29}+\frac{7}{3983323}a^{28}-\frac{23}{3983323}a^{27}-\frac{30}{3983323}a^{26}+\frac{39}{3983323}a^{25}+\frac{60}{3983323}a^{24}-\frac{400398}{3983323}a^{23}-\frac{1528952}{3983323}a^{22}-\frac{103117}{3983323}a^{21}-\frac{506271}{3983323}a^{20}+\frac{273720}{3983323}a^{19}-\frac{815018}{3983323}a^{18}+\frac{1343695}{3983323}a^{17}-\frac{811089}{3983323}a^{16}-\frac{1679061}{3983323}a^{15}-\frac{477397}{3983323}a^{14}+\frac{1538586}{3983323}a^{13}+\frac{952302}{3983323}a^{12}-\frac{1877059}{3983323}a^{11}+\frac{814551}{3983323}a^{10}+\frac{1702555}{3983323}a^{9}-\frac{144244}{3983323}a^{8}+\frac{828752}{3983323}a^{7}-\frac{1090985}{3983323}a^{6}-\frac{1526087}{3983323}a^{5}-\frac{1595416}{3983323}a^{4}+\frac{269515}{3983323}a^{3}+\frac{777450}{3983323}a^{2}-\frac{181133}{3983323}a-\frac{5034}{28657}$, $\frac{1}{3983323}a^{41}+\frac{50}{3983323}a^{39}+\frac{41}{3983323}a^{37}-\frac{58}{3983323}a^{35}-\frac{40}{3983323}a^{33}+\frac{52}{3983323}a^{31}+\frac{63}{3983323}a^{29}-\frac{46}{3983323}a^{27}+\frac{69}{3983323}a^{25}+\frac{783156}{3983323}a^{23}+\frac{9062}{28657}a^{22}+\frac{1219205}{3983323}a^{21}-\frac{7223}{28657}a^{20}+\frac{277101}{3983323}a^{19}+\frac{13744}{28657}a^{18}-\frac{582513}{3983323}a^{17}-\frac{1119}{28657}a^{16}+\frac{1008826}{3983323}a^{15}-\frac{8359}{28657}a^{14}+\frac{230260}{3983323}a^{13}-\frac{9130}{28657}a^{12}+\frac{445655}{3983323}a^{11}-\frac{4052}{28657}a^{10}-\frac{903368}{3983323}a^{9}-\frac{4140}{28657}a^{8}+\frac{839797}{3983323}a^{7}-\frac{11119}{28657}a^{6}+\frac{151417}{3983323}a^{5}+\frac{1695}{28657}a^{4}+\frac{1165558}{3983323}a^{3}-\frac{9371}{28657}a^{2}-\frac{1386657}{3983323}a-\frac{6107}{28657}$, $\frac{1}{5500969063}a^{42}+\frac{114}{5500969063}a^{41}-\frac{154}{5500969063}a^{40}-\frac{31537}{5500969063}a^{39}-\frac{94482}{5500969063}a^{38}-\frac{15074}{5500969063}a^{37}-\frac{68596}{5500969063}a^{36}+\frac{80319}{5500969063}a^{35}-\frac{10632}{5500969063}a^{34}+\frac{5123}{5500969063}a^{33}+\frac{23716}{5500969063}a^{32}-\frac{65590}{5500969063}a^{31}-\frac{43311}{5500969063}a^{30}+\frac{71936}{5500969063}a^{29}+\frac{52319}{5500969063}a^{28}-\frac{55040}{5500969063}a^{27}-\frac{67064}{5500969063}a^{26}+\frac{12698}{5500969063}a^{25}-\frac{95471}{5500969063}a^{24}+\frac{1669779628}{5500969063}a^{23}-\frac{1795757706}{5500969063}a^{22}-\frac{2028011570}{5500969063}a^{21}+\frac{2411464291}{5500969063}a^{20}+\frac{142052597}{5500969063}a^{19}+\frac{1076196489}{5500969063}a^{18}+\frac{2583475676}{5500969063}a^{17}-\frac{860725813}{5500969063}a^{16}-\frac{1504518919}{5500969063}a^{15}-\frac{2517486720}{5500969063}a^{14}-\frac{608195519}{5500969063}a^{13}-\frac{629810102}{5500969063}a^{12}+\frac{2714969632}{5500969063}a^{11}+\frac{2004505629}{5500969063}a^{10}+\frac{2101227756}{5500969063}a^{9}-\frac{269093028}{5500969063}a^{8}+\frac{570400744}{5500969063}a^{7}+\frac{2714670997}{5500969063}a^{6}+\frac{51523621}{5500969063}a^{5}-\frac{650493013}{5500969063}a^{4}-\frac{977972298}{5500969063}a^{3}+\frac{2742876477}{5500969063}a^{2}-\frac{910393118}{5500969063}a-\frac{9053653}{39575317}$, $\frac{1}{15\!\cdots\!19}a^{43}+\frac{66\!\cdots\!52}{15\!\cdots\!19}a^{42}+\frac{13\!\cdots\!34}{15\!\cdots\!19}a^{41}-\frac{31\!\cdots\!91}{15\!\cdots\!19}a^{40}+\frac{18\!\cdots\!20}{15\!\cdots\!19}a^{39}+\frac{10\!\cdots\!05}{15\!\cdots\!19}a^{38}-\frac{73\!\cdots\!94}{15\!\cdots\!19}a^{37}-\frac{10\!\cdots\!86}{15\!\cdots\!19}a^{36}+\frac{23\!\cdots\!96}{15\!\cdots\!19}a^{35}+\frac{36\!\cdots\!34}{15\!\cdots\!19}a^{34}+\frac{12\!\cdots\!82}{15\!\cdots\!19}a^{33}+\frac{19\!\cdots\!12}{15\!\cdots\!19}a^{32}+\frac{11\!\cdots\!40}{15\!\cdots\!19}a^{31}-\frac{49\!\cdots\!27}{15\!\cdots\!19}a^{30}+\frac{56\!\cdots\!40}{15\!\cdots\!19}a^{29}+\frac{26\!\cdots\!91}{15\!\cdots\!19}a^{28}-\frac{70\!\cdots\!19}{15\!\cdots\!19}a^{27}+\frac{24\!\cdots\!38}{15\!\cdots\!19}a^{26}-\frac{10\!\cdots\!50}{15\!\cdots\!19}a^{25}-\frac{70\!\cdots\!92}{15\!\cdots\!19}a^{24}-\frac{13\!\cdots\!09}{15\!\cdots\!19}a^{23}+\frac{65\!\cdots\!78}{15\!\cdots\!19}a^{22}+\frac{27\!\cdots\!41}{15\!\cdots\!19}a^{21}-\frac{47\!\cdots\!29}{15\!\cdots\!19}a^{20}-\frac{60\!\cdots\!73}{15\!\cdots\!19}a^{19}-\frac{69\!\cdots\!92}{15\!\cdots\!19}a^{18}+\frac{19\!\cdots\!20}{15\!\cdots\!19}a^{17}-\frac{19\!\cdots\!56}{15\!\cdots\!19}a^{16}+\frac{51\!\cdots\!88}{15\!\cdots\!19}a^{15}+\frac{13\!\cdots\!52}{15\!\cdots\!19}a^{14}+\frac{66\!\cdots\!44}{15\!\cdots\!19}a^{13}+\frac{39\!\cdots\!35}{15\!\cdots\!19}a^{12}-\frac{42\!\cdots\!02}{15\!\cdots\!19}a^{11}-\frac{47\!\cdots\!20}{15\!\cdots\!19}a^{10}-\frac{23\!\cdots\!62}{15\!\cdots\!19}a^{9}+\frac{56\!\cdots\!17}{15\!\cdots\!19}a^{8}-\frac{72\!\cdots\!52}{15\!\cdots\!19}a^{7}-\frac{44\!\cdots\!75}{15\!\cdots\!19}a^{6}-\frac{58\!\cdots\!56}{15\!\cdots\!19}a^{5}-\frac{76\!\cdots\!03}{15\!\cdots\!19}a^{4}-\frac{61\!\cdots\!77}{15\!\cdots\!19}a^{3}-\frac{63\!\cdots\!63}{15\!\cdots\!19}a^{2}+\frac{57\!\cdots\!44}{15\!\cdots\!19}a+\frac{34\!\cdots\!34}{24\!\cdots\!61}$ 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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{30396202823626472809883862808913371608380021398424097524446400111803106595028279803557138639319633495530237165205872335129972}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{43} - \frac{16244059232468368819987914929694039806780558468851032754244454209648431429251629725845218144259362961200440049337307929888800}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{42} + \frac{968137064383691479455797790027366347478663284438929571650112883475526246919001940066012149891662258369411027161661740874094087}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{41} + \frac{284122481947603482556560960554884176187921625732525718680452049622648738230216359699266612936093171283406637978849786878228258}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{40} - \frac{14883588320661136659929700316490985428854579067225418325980625740019895978144572184835543089775237610359235308951150049410522859}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{39} - \frac{2462743070832765499617484195363019906262497395764846070892184381067966960650020706656551067152279969688802486999936456092964351}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{38} + \frac{1104575138588386186827193405875070672891618309240865751615711524130564373169868176855600885141759848674152108363983453498961016}{3903492581300256032647597157355847220156091733271915924671863199904016501374601659191406141506673521529530825400014401377173166098553} a^{37} + \frac{9571995544119863113726595260203225772324307488523262391189013931447753615447919392396936867407206613996299905554999844319760564}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{36} - \frac{1169970746852433326510935455880975723698005903073657939567948854056750930053535537166329904931749145569333888278024177117509265252}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{35} + \frac{31670898463367378090005780259162813051189868217223692691344080344872159253480862427039321527039904724070242911635551034163561209}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{34} + \frac{6996152673033287308700272995699466578326382458540889679724224204348258400144740567499402240207644002960359231160365230874935595571}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{33} - \frac{772437944830730513728819439271512979672367038243367244593188527328440724072927186150439019520472883197531142313657787875496139637}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{32} - \frac{33751144127957791376408336763214066199943378956596995109694931442343538641956560213269703544691198131854774188773656989706994314144}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{31} + \frac{6490338193883388350265965265857157785810555023445301815181929558856092572960658263361951407218833847207084096897211952106255862350}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{30} + \frac{134216350151836949377167596351755171737354381595125831992299036627946432241054240247154408978208082470670414546235712828676482374079}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{29} - \frac{36898831737542735710660465127603559239833584050939901625652609577704930438972880323184008551536056021210908595968056324466852038138}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{28} - \frac{443968286639389537687370538990850854941011795436105648613314885021585642208080929985591060644051470513715432789549536144845246629026}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{27} + \frac{160726411830574013792284855229708953347044942619661139309527621850180982725173067621408920838913949230131789272371856538871088441057}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{26} + \frac{1231933711184688417933492721837288724461445020220142242305151823444742019543667762620235057667050551853290560863749463704130852747875}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{25} - \frac{559909475811860824918017126765600716557920246969181757270720922850269024976144405888986980053518222548616488607626206070646765096040}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{24} - \frac{2868372626328137287741484936525240045737558568718016880276543514547368194996568054739630323321581826847280822510674240193600539687383}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{23} + \frac{1597012828417528991319614848312256354444111305822888928851514712055463031887532253632904577608088809194476347074868853262338092735613}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{22} + \frac{5624245425351843116992250953981742542442663811751689706605543680232778442369209766466230897925459316806040196185237973186167222380399}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{21} - \frac{3750467672988327205397646287351771740715980756666237767773197644018820936909966645334633714429796540793806152795295034491363470163978}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{20} - \frac{9254132208484057015290670213101617197631380875258143405485413283775368899291077259149452678474569365585604269853132632187960788580301}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{19} + \frac{7388166564284404154920579974771904248766662068553483294119660434590435792826029538443485541862338196573890655813377708325805588765174}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{18} + \frac{12985575172140954063708974669388355868004903096367656276893736372401966157132866574592995343920431340060727738097189364649746216263721}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{17} - \frac{12747746011507601680484551317941272004722197828793057556326359015640557764248399796756957815976427250696472181277859584546970687792603}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{16} - \frac{15506221765282796773733162148250114323158253476242987300888111561135837315442735260693992388644830590615550400610247771031190198854296}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{15} + \frac{15229536539553680463773706482699717647302704328176202426392833225450979822163671005991692011054238179087716909870031097983190791318820}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{14} + \frac{13922693004865941537839117779306028367678183787869340853522632851762262835905254719758966772149651396265801938489430795657012221834105}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{13} + \frac{29094700593615157317413103488766831379359986909155660694482678736300245006766123308764878637604319961054370995371425169742428909778985}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{12} - \frac{14131180285463567132638421687239915761091284302438857022198871718933769093115175039461099364872061415336967970361830876304247160904735}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{11} - \frac{130829983360154711532651293095460741229664091380445371146933513828650748909552601688795122817977293397727606392269100996657682424746686}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{10} + \frac{100086180798998278912854405561416875111966005683184860111957476119693861203424705490420934211564073833168087477430371919012130902355787}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{9} + \frac{23177173636597789063224518273870427668865499690994442966484673752878068206617690528206198912511356252697998130643697107232718136431335}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{8} - \frac{235576122257433666413887440259554852617359443937000754859662265714092516605708456857127358935014352923469748574082351467374623225266679}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{7} + \frac{327931366952256047738672216783892350953897830632542422738668819415741268520060454029566605695532979998135378303103112169782099131034938}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{6} + \frac{18836082930136174628636166387263271362527159077828686429710244491002347579984769108318817394798930828919986193283143698175872185246745}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{5} - \frac{438423885413599488414134181834383024435638706364497351381923530629376031522198619928034146620074586512323331916792344961812018480407259}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{4} + \frac{386886777460609735767238860245816411646468192718569229105292590785449723458023611276698815968243196931662995846780928056271625705779273}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{3} - \frac{96628510794923113424672257609959036731096430588468088795150846541105134619079078059603977821041921555150538186797206610715500793257744}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a^{2} - \frac{513093405889685415836534827638314283809859316346556566268285086476198000973543445336233066970142955379745039964017055465268694534211811}{542585468800735588538016004872462763601696750924796313529388984786658293691069630627605453669427619492604784730602001791427070087698867} a + \frac{10517775915641457695712169198631401228854761233558600729390777724711416516133785463229438130567557111588750896862676065546352633660}{8467445946421379680363551317474722820295209833561639749830505856624764645064211842063787725611005469695294632104152714484106650973} \)  (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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241)
 
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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241, 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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);
 
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 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 4106118241);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{345}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{-3}, \sqrt{-115})\), \(\Q(\zeta_{23})^+\), 22.22.341419566026798986253349758444608447265625.1, 22.0.304011857053427966889939263171547.1, 22.0.1927323443393334271838358868310546875.1

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 R ${\href{/padicField/7.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ ${\href{/padicField/37.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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$
\(5\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$