Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536)
gp: K = bnfinit(y^21 - 84*y^19 + 3024*y^17 - 60928*y^15 - 138*y^14 + 752640*y^13 + 7728*y^12 - 5870592*y^11 - 170016*y^10 + 28700672*y^9 + 1854720*y^8 - 84318144*y^7 - 10386432*y^6 + 135502080*y^5 + 27697152*y^4 - 94947328*y^3 - 27697152*y^2 + 10063872*y - 577536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536)
\( x^{21} - 84 x^{19} + 3024 x^{17} - 60928 x^{15} - 138 x^{14} + 752640 x^{13} + 7728 x^{12} + \cdots - 577536 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6623268093347351492485520944725602280711389184\)
\(\medspace = 2^{18}\cdot 3^{28}\cdot 7^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(152.04\) | 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: | $2^{6/7}3^{4/3}7^{236/147}\approx 178.21465760874582$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14}a^{7}-\frac{2}{7}$, $\frac{1}{14}a^{8}-\frac{2}{7}a$, $\frac{1}{28}a^{9}+\frac{5}{14}a^{2}$, $\frac{1}{56}a^{10}-\frac{9}{28}a^{3}$, $\frac{1}{56}a^{11}-\frac{9}{28}a^{4}$, $\frac{1}{784}a^{12}-\frac{3}{392}a^{11}+\frac{3}{392}a^{10}+\frac{1}{196}a^{9}-\frac{1}{98}a^{8}-\frac{1}{49}a^{7}+\frac{3}{7}a^{6}+\frac{131}{392}a^{5}+\frac{27}{196}a^{4}-\frac{69}{196}a^{3}+\frac{19}{98}a^{2}-\frac{5}{49}a-\frac{24}{49}$, $\frac{1}{1568}a^{13}+\frac{3}{392}a^{11}+\frac{3}{392}a^{10}+\frac{1}{98}a^{9}+\frac{3}{98}a^{8}+\frac{1}{98}a^{7}-\frac{37}{784}a^{6}-\frac{3}{7}a^{5}+\frac{25}{98}a^{4}-\frac{27}{196}a^{3}-\frac{23}{49}a^{2}+\frac{8}{49}a+\frac{5}{49}$, $\frac{1}{3136}a^{14}-\frac{53}{1568}a^{7}-\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{12}{49}$, $\frac{1}{3136}a^{15}-\frac{53}{1568}a^{8}-\frac{3}{8}a^{6}-\frac{1}{2}a^{4}-\frac{12}{49}a$, $\frac{1}{6272}a^{16}-\frac{53}{3136}a^{9}+\frac{3}{112}a^{7}+\frac{1}{4}a^{5}-\frac{6}{49}a^{2}+\frac{1}{7}$, $\frac{1}{87808}a^{17}+\frac{1}{43904}a^{16}-\frac{1}{10976}a^{15}-\frac{1}{10976}a^{14}+\frac{3}{392}a^{11}+\frac{59}{43904}a^{10}-\frac{389}{21952}a^{9}-\frac{377}{10976}a^{8}+\frac{37}{2744}a^{7}+\frac{1}{8}a^{6}+\frac{2}{7}a^{5}+\frac{1}{196}a^{4}-\frac{271}{1372}a^{3}+\frac{275}{686}a^{2}+\frac{164}{343}a+\frac{129}{343}$, $\frac{1}{12468736}a^{18}+\frac{1}{6234368}a^{17}+\frac{31}{779296}a^{16}+\frac{125}{1558592}a^{15}-\frac{1}{111328}a^{14}+\frac{17}{55664}a^{13}-\frac{1}{3976}a^{12}+\frac{37467}{6234368}a^{11}-\frac{26821}{3117184}a^{10}+\frac{4929}{1558592}a^{9}-\frac{1945}{194824}a^{8}-\frac{62}{3479}a^{7}-\frac{7447}{27832}a^{6}-\frac{19}{71}a^{5}+\frac{74335}{194824}a^{4}+\frac{16025}{97412}a^{3}-\frac{16027}{48706}a^{2}-\frac{737}{24353}a+\frac{29}{497}$, $\frac{1}{24937472}a^{19}-\frac{19}{6234368}a^{17}-\frac{141}{3117184}a^{16}+\frac{19}{194824}a^{15}+\frac{39}{1558592}a^{14}+\frac{23}{111328}a^{13}-\frac{7109}{12468736}a^{12}+\frac{139}{27832}a^{11}-\frac{6085}{779296}a^{10}-\frac{297}{21952}a^{9}-\frac{4623}{389648}a^{8}+\frac{26283}{779296}a^{7}-\frac{25001}{55664}a^{6}+\frac{128907}{389648}a^{5}+\frac{225}{6958}a^{4}-\frac{761}{48706}a^{3}-\frac{19287}{48706}a^{2}-\frac{8670}{24353}a-\frac{6817}{24353}$, $\frac{1}{49874944}a^{20}+\frac{5}{1558592}a^{17}+\frac{165}{3117184}a^{16}-\frac{9}{111328}a^{15}+\frac{27}{389648}a^{14}-\frac{5541}{24937472}a^{13}+\frac{15}{55664}a^{12}-\frac{2885}{890624}a^{11}-\frac{5333}{779296}a^{10}-\frac{2753}{194824}a^{9}+\frac{419}{55664}a^{8}+\frac{8009}{389648}a^{7}-\frac{155097}{779296}a^{6}+\frac{11233}{27832}a^{5}-\frac{748}{3479}a^{4}-\frac{28135}{97412}a^{3}+\frac{5821}{24353}a^{2}-\frac{1397}{3479}a-\frac{10347}{24353}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
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: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{101}{24937472}a^{20}-\frac{145}{24937472}a^{19}-\frac{4097}{12468736}a^{18}+\frac{5}{10976}a^{17}+\frac{2531}{222656}a^{16}-\frac{11819}{779296}a^{15}-\frac{170383}{779296}a^{14}+\frac{3472695}{12468736}a^{13}+\frac{31861269}{12468736}a^{12}-\frac{19055833}{6234368}a^{11}-\frac{58068597}{3117184}a^{10}+\frac{4503997}{222656}a^{9}+\frac{65254661}{779296}a^{8}-\frac{3749097}{48706}a^{7}-\frac{43187211}{194824}a^{6}+\frac{57449627}{389648}a^{5}+\frac{61349389}{194824}a^{4}-\frac{9323717}{97412}a^{3}-\frac{1319587}{6958}a^{2}-\frac{703652}{24353}a+\frac{198473}{24353}$, $\frac{3}{194824}a^{20}-\frac{171}{24937472}a^{19}-\frac{15041}{12468736}a^{18}+\frac{2787}{6234368}a^{17}+\frac{124363}{3117184}a^{16}-\frac{17299}{1558592}a^{15}-\frac{141067}{194824}a^{14}+\frac{91597}{779296}a^{13}+\frac{97817879}{12468736}a^{12}-\frac{793787}{6234368}a^{11}-\frac{80523543}{1558592}a^{10}-\frac{3341913}{389648}a^{9}+\frac{156483867}{779296}a^{8}+\frac{15065333}{194824}a^{7}-\frac{162950069}{389648}a^{6}-\frac{102905655}{389648}a^{5}+\frac{68166893}{194824}a^{4}+\frac{31220751}{97412}a^{3}+\frac{170297}{24353}a^{2}-\frac{700585}{24353}a+\frac{50345}{24353}$, $\frac{57}{24937472}a^{19}+\frac{153}{12468736}a^{18}-\frac{1143}{6234368}a^{17}-\frac{2697}{3117184}a^{16}+\frac{4913}{779296}a^{15}+\frac{19885}{779296}a^{14}-\frac{6719}{55664}a^{13}-\frac{5099581}{12468736}a^{12}+\frac{8715907}{6234368}a^{11}+\frac{1510783}{389648}a^{10}-\frac{7762975}{779296}a^{9}-\frac{4320531}{194824}a^{8}+\frac{16538803}{389648}a^{7}+\frac{1038841}{13916}a^{6}-\frac{38749961}{389648}a^{5}-\frac{26416371}{194824}a^{4}+\frac{5093961}{48706}a^{3}+\frac{4922431}{48706}a^{2}-\frac{570315}{24353}a+\frac{28993}{24353}$, $\frac{3}{3117184}a^{20}+\frac{305}{24937472}a^{19}-\frac{347}{6234368}a^{18}-\frac{5387}{6234368}a^{17}+\frac{3251}{3117184}a^{16}+\frac{39259}{1558592}a^{15}-\frac{909}{779296}a^{14}-\frac{608585}{1558592}a^{13}-\frac{2820533}{12468736}a^{12}+\frac{10760815}{3117184}a^{11}+\frac{2728513}{779296}a^{10}-\frac{26861693}{1558592}a^{9}-\frac{18965749}{779296}a^{8}+\frac{8559409}{194824}a^{7}+\frac{33914957}{389648}a^{6}-\frac{14511727}{389648}a^{5}-\frac{14833239}{97412}a^{4}-\frac{3572773}{97412}a^{3}+\frac{2593748}{24353}a^{2}+\frac{1327941}{24353}a-\frac{260515}{24353}$, $\frac{45}{6234368}a^{20}+\frac{39}{6234368}a^{19}-\frac{835}{1558592}a^{18}-\frac{2657}{6234368}a^{17}+\frac{6535}{389648}a^{16}+\frac{9249}{779296}a^{15}-\frac{14017}{48706}a^{14}-\frac{542253}{3117184}a^{13}+\frac{9170677}{3117184}a^{12}+\frac{70249}{48706}a^{11}-\frac{56738239}{3117184}a^{10}-\frac{1327181}{194824}a^{9}+\frac{51376197}{779296}a^{8}+\frac{438177}{24353}a^{7}-\frac{49511179}{389648}a^{6}-\frac{5160181}{194824}a^{5}+\frac{10217275}{97412}a^{4}+\frac{851763}{48706}a^{3}-\frac{479279}{24353}a^{2}+\frac{232457}{24353}a+\frac{138667}{24353}$, $\frac{1}{222656}a^{20}-\frac{299}{24937472}a^{19}-\frac{4633}{12468736}a^{18}+\frac{381}{389648}a^{17}+\frac{40961}{3117184}a^{16}-\frac{7509}{222656}a^{15}-\frac{100837}{389648}a^{14}+\frac{71105}{111328}a^{13}+\frac{38623255}{12468736}a^{12}-\frac{44949923}{6234368}a^{11}-\frac{71941887}{3117184}a^{10}+\frac{38348883}{779296}a^{9}+\frac{2935887}{27832}a^{8}-\frac{38220741}{194824}a^{7}-\frac{3922867}{13916}a^{6}+\frac{162002817}{389648}a^{5}+\frac{78515551}{194824}a^{4}-\frac{37068537}{97412}a^{3}-\frac{11852343}{48706}a^{2}+\frac{250290}{3479}a-\frac{104519}{24353}$, $\frac{167}{49874944}a^{20}-\frac{527}{24937472}a^{19}-\frac{3397}{12468736}a^{18}+\frac{9743}{6234368}a^{17}+\frac{14409}{1558592}a^{16}-\frac{76057}{1558592}a^{15}-\frac{66041}{389648}a^{14}+\frac{20855965}{24937472}a^{13}+\frac{22642187}{12468736}a^{12}-\frac{13329615}{1558592}a^{11}-\frac{17859735}{1558592}a^{10}+\frac{20670651}{389648}a^{9}+\frac{31609761}{779296}a^{8}-\frac{75499519}{389648}a^{7}-\frac{54895899}{779296}a^{6}+\frac{149723549}{389648}a^{5}+\frac{7994843}{194824}a^{4}-\frac{8582921}{24353}a^{3}+\frac{380679}{48706}a^{2}+\frac{2606544}{24353}a-\frac{380257}{24353}$, $\frac{821}{49874944}a^{20}+\frac{219}{24937472}a^{19}-\frac{4349}{3117184}a^{18}-\frac{43}{63616}a^{17}+\frac{809}{15904}a^{16}+\frac{8433}{389648}a^{15}-\frac{811873}{779296}a^{14}-\frac{9392489}{24937472}a^{13}+\frac{163632377}{12468736}a^{12}+\frac{24561317}{6234368}a^{11}-\frac{46575525}{445312}a^{10}-\frac{5862037}{222656}a^{9}+\frac{407040729}{779296}a^{8}+\frac{46483307}{389648}a^{7}-\frac{1216807329}{779296}a^{6}-\frac{146937661}{389648}a^{5}+\frac{246476271}{97412}a^{4}+\frac{5151557}{6958}a^{3}-\frac{12147829}{6958}a^{2}-\frac{15733713}{24353}a+\frac{3300833}{24353}$, $\frac{281}{49874944}a^{20}+\frac{325}{24937472}a^{19}-\frac{5227}{12468736}a^{18}-\frac{6137}{6234368}a^{17}+\frac{40879}{3117184}a^{16}+\frac{48183}{1558592}a^{15}-\frac{87443}{389648}a^{14}-\frac{13052253}{24937472}a^{13}+\frac{28575815}{12468736}a^{12}+\frac{8081491}{1558592}a^{11}-\frac{22310425}{1558592}a^{10}-\frac{671303}{21952}a^{9}+\frac{42012781}{779296}a^{8}+\frac{10210513}{97412}a^{7}-\frac{90469349}{779296}a^{6}-\frac{76071129}{389648}a^{5}+\frac{24961199}{194824}a^{4}+\frac{16714317}{97412}a^{3}-\frac{2735861}{48706}a^{2}-\frac{1202505}{24353}a+\frac{2269}{343}$, $\frac{689}{49874944}a^{20}-\frac{711}{12468736}a^{19}-\frac{13843}{12468736}a^{18}+\frac{849}{194824}a^{17}+\frac{58847}{1558592}a^{16}-\frac{15587}{111328}a^{15}-\frac{549875}{779296}a^{14}+\frac{61170923}{24937472}a^{13}+\frac{49094019}{6234368}a^{12}-\frac{39598821}{1558592}a^{11}-\frac{83299771}{1558592}a^{10}+\frac{61373589}{389648}a^{9}+\frac{6005343}{27832}a^{8}-\frac{13705621}{24353}a^{7}-\frac{381958585}{779296}a^{6}+\frac{203723547}{194824}a^{5}+\frac{112692257}{194824}a^{4}-\frac{19366938}{24353}a^{3}-\frac{7597979}{24353}a^{2}+\frac{353761}{3479}a-\frac{135011}{24353}$, $\frac{633}{49874944}a^{20}-\frac{279}{12468736}a^{19}-\frac{13035}{12468736}a^{18}+\frac{5575}{3117184}a^{17}+\frac{57185}{1558592}a^{16}-\frac{23641}{389648}a^{15}-\frac{278415}{389648}a^{14}+\frac{28302563}{24937472}a^{13}+\frac{52518483}{6234368}a^{12}-\frac{19817797}{1558592}a^{11}-\frac{47958539}{779296}a^{10}+\frac{8435685}{97412}a^{9}+\frac{53388455}{194824}a^{8}-\frac{135090861}{389648}a^{7}-\frac{548927303}{779296}a^{6}+\frac{144915409}{194824}a^{5}+\frac{182840005}{194824}a^{4}-\frac{67903075}{97412}a^{3}-\frac{12067002}{24353}a^{2}+\frac{3390693}{24353}a-\frac{181997}{24353}$, $\frac{145}{49874944}a^{20}-\frac{145}{24937472}a^{19}-\frac{1339}{6234368}a^{18}+\frac{691}{1558592}a^{17}+\frac{10361}{1558592}a^{16}-\frac{22261}{1558592}a^{15}-\frac{43559}{389648}a^{14}+\frac{6299371}{24937472}a^{13}+\frac{13812693}{12468736}a^{12}-\frac{16617893}{6234368}a^{11}-\frac{20478041}{3117184}a^{10}+\frac{26578201}{1558592}a^{9}+\frac{4435565}{194824}a^{8}-\frac{3120791}{48706}a^{7}-\frac{34481787}{779296}a^{6}+\frac{50244975}{389648}a^{5}+\frac{4766135}{97412}a^{4}-\frac{10824263}{97412}a^{3}-\frac{762284}{24353}a^{2}+\frac{425612}{24353}a-\frac{27823}{24353}$, $\frac{27}{3562496}a^{20}+\frac{51}{24937472}a^{19}-\frac{7789}{12468736}a^{18}-\frac{23}{127232}a^{17}+\frac{8563}{389648}a^{16}+\frac{10177}{1558592}a^{15}-\frac{335409}{779296}a^{14}-\frac{223927}{1781248}a^{13}+\frac{64012353}{12468736}a^{12}+\frac{8784691}{6234368}a^{11}-\frac{120307}{3136}a^{10}-\frac{14784519}{1558592}a^{9}+\frac{138809731}{779296}a^{8}+\frac{3803111}{97412}a^{7}-\frac{13701377}{27832}a^{6}-\frac{39927121}{389648}a^{5}+\frac{142879293}{194824}a^{4}+\frac{612310}{3479}a^{3}-\frac{22058711}{48706}a^{2}-\frac{3811747}{24353}a+\frac{367477}{24353}$, $\frac{11}{24937472}a^{20}-\frac{53}{1781248}a^{19}-\frac{507}{12468736}a^{18}+\frac{13463}{6234368}a^{17}+\frac{2031}{1558592}a^{16}-\frac{102251}{1558592}a^{15}-\frac{13637}{779296}a^{14}+\frac{13488025}{12468736}a^{13}+\frac{49177}{890624}a^{12}-\frac{65444731}{6234368}a^{11}+\frac{3093103}{3117184}a^{10}+\frac{94850209}{1558592}a^{9}-\frac{2090513}{194824}a^{8}-\frac{9990137}{48706}a^{7}+\frac{7578771}{194824}a^{6}+\frac{1302622}{3479}a^{5}-\frac{10133087}{194824}a^{4}-\frac{15260845}{48706}a^{3}+\frac{678507}{48706}a^{2}+\frac{1758620}{24353}a-\frac{146885}{24353}$, $\frac{909}{24937472}a^{20}+\frac{575}{12468736}a^{19}-\frac{36669}{12468736}a^{18}-\frac{1487}{389648}a^{17}+\frac{313713}{3117184}a^{16}+\frac{104319}{779296}a^{15}-\frac{1479867}{779296}a^{14}-\frac{32306305}{12468736}a^{13}+\frac{133868325}{6234368}a^{12}+\frac{188005123}{6234368}a^{11}-\frac{230053693}{1558592}a^{10}-\frac{83864817}{389648}a^{9}+\frac{57737409}{97412}a^{8}+\frac{356758691}{389648}a^{7}-\frac{477182407}{389648}a^{6}-\frac{411542983}{194824}a^{5}+\frac{162734797}{194824}a^{4}+\frac{198545475}{97412}a^{3}+\frac{18812649}{48706}a^{2}-\frac{4206574}{24353}a+\frac{246643}{24353}$, $\frac{79}{49874944}a^{20}+\frac{3}{3117184}a^{19}-\frac{1825}{12468736}a^{18}-\frac{701}{6234368}a^{17}+\frac{144}{24353}a^{16}+\frac{951}{194824}a^{15}-\frac{105747}{779296}a^{14}-\frac{2714699}{24937472}a^{13}+\frac{42303}{21952}a^{12}+\frac{1090011}{779296}a^{11}-\frac{54041095}{3117184}a^{10}-\frac{8423039}{779296}a^{9}+\frac{75883623}{779296}a^{8}+\frac{1232661}{24353}a^{7}-\frac{252662805}{779296}a^{6}-\frac{13809343}{97412}a^{5}+\frac{111819761}{194824}a^{4}+\frac{11096525}{48706}a^{3}-\frac{20485925}{48706}a^{2}-\frac{4201535}{24353}a+\frac{861379}{24353}$, $\frac{257}{24937472}a^{20}-\frac{39}{6234368}a^{19}-\frac{5315}{6234368}a^{18}+\frac{3111}{6234368}a^{17}+\frac{5863}{194824}a^{16}-\frac{6655}{389648}a^{15}-\frac{230249}{389648}a^{14}+\frac{4055227}{12468736}a^{13}+\frac{21980835}{3117184}a^{12}-\frac{2899461}{779296}a^{11}-\frac{163494287}{3117184}a^{10}+\frac{9982797}{389648}a^{9}+\frac{186983835}{779296}a^{8}-\frac{2428385}{24353}a^{7}-\frac{250410857}{389648}a^{6}+\frac{17860611}{97412}a^{5}+\frac{88937215}{97412}a^{4}-\frac{7032859}{97412}a^{3}-\frac{26297339}{48706}a^{2}-\frac{2766265}{24353}a+\frac{725125}{24353}$, $\frac{19}{1781248}a^{20}+\frac{1}{3117184}a^{19}-\frac{9897}{12468736}a^{18}-\frac{663}{3117184}a^{17}+\frac{77145}{3117184}a^{16}+\frac{9717}{779296}a^{15}-\frac{325849}{779296}a^{14}-\frac{275575}{890624}a^{13}+\frac{6431855}{1558592}a^{12}+\frac{25312017}{6234368}a^{11}-\frac{37058841}{1558592}a^{10}-\frac{23272903}{779296}a^{9}+\frac{3612143}{48706}a^{8}+\frac{46994993}{389648}a^{7}-\frac{5311083}{55664}a^{6}-\frac{46858097}{194824}a^{5}-\frac{5564935}{194824}a^{4}+\frac{4216165}{24353}a^{3}+\frac{6007263}{48706}a^{2}+\frac{596621}{24353}a-\frac{50207}{24353}$, $\frac{3}{194824}a^{20}-\frac{13}{508928}a^{19}-\frac{1097}{890624}a^{18}+\frac{13099}{6234368}a^{17}+\frac{130171}{3117184}a^{16}-\frac{2043}{27832}a^{15}-\frac{607701}{779296}a^{14}+\frac{68727}{48706}a^{13}+\frac{15529863}{1781248}a^{12}-\frac{7187491}{445312}a^{11}-\frac{2890511}{48706}a^{10}+\frac{172577731}{1558592}a^{9}+\frac{13314347}{55664}a^{8}-\frac{84783819}{194824}a^{7}-\frac{103701711}{194824}a^{6}+\frac{48556285}{55664}a^{5}+\frac{4127777}{6958}a^{4}-\frac{67657383}{97412}a^{3}-\frac{7200068}{24353}a^{2}+\frac{317965}{3479}a-\frac{123527}{24353}$, $\frac{9}{49874944}a^{20}+\frac{283}{24937472}a^{19}-\frac{397}{6234368}a^{18}-\frac{4003}{6234368}a^{17}+\frac{351}{97412}a^{16}+\frac{149}{10976}a^{15}-\frac{70691}{779296}a^{14}-\frac{2905421}{24937472}a^{13}+\frac{15633081}{12468736}a^{12}-\frac{284825}{6234368}a^{11}-\frac{16009625}{1558592}a^{10}+\frac{12908775}{1558592}a^{9}+\frac{19920907}{389648}a^{8}-\frac{23836541}{389648}a^{7}-\frac{119053469}{779296}a^{6}+\frac{73041611}{389648}a^{5}+\frac{12368127}{48706}a^{4}-\frac{11046433}{48706}a^{3}-\frac{8606555}{48706}a^{2}+\frac{1203980}{24353}a-\frac{65729}{24353}$
| 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: | \( 72918598906000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 72918598906000000 \cdot 3}{2\cdot\sqrt{6623268093347351492485520944725602280711389184}}\cr\approx \mathstrut & 2.81853424628225 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536)
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^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536, 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^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536);
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^21 - 84*x^19 + 3024*x^17 - 60928*x^15 - 138*x^14 + 752640*x^13 + 7728*x^12 - 5870592*x^11 - 170016*x^10 + 28700672*x^9 + 1854720*x^8 - 84318144*x^7 - 10386432*x^6 + 135502080*x^5 + 27697152*x^4 - 94947328*x^3 - 27697152*x^2 + 10063872*x - 577536);
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
$C_7^2:C_3^2$ (as 21T21):
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)
| A solvable group of order 441 |
| The 25 conjugacy class representatives for $C_7^2:C_3^2$ |
| Character table for $C_7^2:C_3^2$ |
Intermediate fields
| 3.3.3969.1 |
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]
Sibling fields
| Degree 21 sibling: | data not computed |
| Minimal sibling: | 21.21.103488563958552367070086264761337535636115456.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $21$ | R | $21$ | $21$ | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(3\)
| 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
| 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
|
\(7\)
| Deg $21$ | $21$ | $1$ | $32$ |