Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405)
gp: K = bnfinit(y^21 - 7*y^20 + 35*y^19 - 112*y^18 + 266*y^17 - 490*y^16 + 693*y^15 - 1355*y^14 + 3360*y^13 - 7224*y^12 + 12166*y^11 - 16709*y^10 + 21700*y^9 - 32417*y^8 + 39987*y^7 - 39067*y^6 + 26600*y^5 - 12320*y^4 + 4389*y^3 - 1575*y^2 + 945*y - 405, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405)
\( x^{21} - 7 x^{20} + 35 x^{19} - 112 x^{18} + 266 x^{17} - 490 x^{16} + 693 x^{15} - 1355 x^{14} + \cdots - 405 \)
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: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2807535013090453863383795166015625\)
\(\medspace = 5^{14}\cdot 7^{28}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.15\) | 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: | $5^{2/3}7^{19/14}\approx 41.01085007711686$ | ||
| Ramified primes: |
\(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{63}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{3}a^{8}+\frac{1}{7}a^{7}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{2}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a-\frac{3}{7}$, $\frac{1}{315}a^{15}+\frac{1}{15}a^{13}+\frac{1}{45}a^{12}-\frac{1}{45}a^{11}-\frac{1}{45}a^{10}+\frac{52}{105}a^{8}-\frac{19}{45}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{16}{45}a^{4}+\frac{19}{45}a^{3}+\frac{2}{9}a^{2}-\frac{2}{7}a$, $\frac{1}{315}a^{16}+\frac{1}{315}a^{14}-\frac{4}{45}a^{13}-\frac{2}{15}a^{12}+\frac{4}{45}a^{11}-\frac{1}{9}a^{10}+\frac{17}{105}a^{9}-\frac{4}{45}a^{8}+\frac{2}{21}a^{7}+\frac{1}{9}a^{6}+\frac{4}{45}a^{5}+\frac{1}{5}a^{4}+\frac{4}{9}a^{3}+\frac{10}{63}a^{2}-\frac{1}{3}a-\frac{2}{7}$, $\frac{1}{315}a^{17}+\frac{2}{315}a^{14}+\frac{2}{15}a^{13}+\frac{1}{15}a^{12}-\frac{4}{45}a^{11}-\frac{47}{315}a^{10}-\frac{4}{45}a^{9}-\frac{2}{5}a^{8}+\frac{41}{105}a^{7}-\frac{11}{45}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{83}{315}a^{3}-\frac{2}{9}a^{2}+\frac{3}{7}$, $\frac{1}{945}a^{18}+\frac{1}{945}a^{17}-\frac{1}{945}a^{16}+\frac{1}{945}a^{15}+\frac{1}{315}a^{14}-\frac{1}{9}a^{13}+\frac{1}{15}a^{12}-\frac{26}{945}a^{11}-\frac{103}{945}a^{10}+\frac{22}{189}a^{9}-\frac{236}{945}a^{8}+\frac{419}{945}a^{7}+\frac{1}{45}a^{6}+\frac{7}{45}a^{5}+\frac{32}{105}a^{4}+\frac{344}{945}a^{3}+\frac{2}{21}a^{2}+\frac{5}{21}a-\frac{2}{7}$, $\frac{1}{208845}a^{19}-\frac{2}{4641}a^{18}-\frac{311}{208845}a^{17}+\frac{62}{208845}a^{16}+\frac{40}{41769}a^{15}+\frac{146}{23205}a^{14}-\frac{304}{3315}a^{13}+\frac{34498}{208845}a^{12}-\frac{4541}{208845}a^{11}-\frac{31}{273}a^{10}+\frac{17666}{208845}a^{9}+\frac{69742}{208845}a^{8}+\frac{101356}{208845}a^{7}-\frac{44}{255}a^{6}+\frac{5248}{69615}a^{5}+\frac{87977}{208845}a^{4}+\frac{44377}{208845}a^{3}-\frac{4187}{13923}a^{2}-\frac{25}{91}a+\frac{90}{1547}$, $\frac{1}{23\!\cdots\!65}a^{20}+\frac{25\!\cdots\!92}{12\!\cdots\!35}a^{19}+\frac{18\!\cdots\!54}{50\!\cdots\!95}a^{18}+\frac{33\!\cdots\!49}{12\!\cdots\!35}a^{17}+\frac{38\!\cdots\!20}{25\!\cdots\!67}a^{16}-\frac{45\!\cdots\!38}{34\!\cdots\!95}a^{15}+\frac{50\!\cdots\!02}{79\!\cdots\!55}a^{14}-\frac{16\!\cdots\!13}{11\!\cdots\!53}a^{13}+\frac{14\!\cdots\!57}{88\!\cdots\!95}a^{12}-\frac{57\!\cdots\!86}{61\!\cdots\!35}a^{11}+\frac{19\!\cdots\!66}{23\!\cdots\!65}a^{10}-\frac{44\!\cdots\!11}{23\!\cdots\!65}a^{9}+\frac{10\!\cdots\!41}{34\!\cdots\!95}a^{8}-\frac{37\!\cdots\!83}{14\!\cdots\!45}a^{7}+\frac{35\!\cdots\!33}{79\!\cdots\!55}a^{6}+\frac{47\!\cdots\!21}{47\!\cdots\!73}a^{5}+\frac{37\!\cdots\!23}{23\!\cdots\!65}a^{4}+\frac{83\!\cdots\!44}{12\!\cdots\!35}a^{3}-\frac{58\!\cdots\!94}{20\!\cdots\!29}a^{2}+\frac{28\!\cdots\!64}{75\!\cdots\!71}a+\frac{71\!\cdots\!59}{17\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (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: | $10$ | 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: |
$\frac{33\!\cdots\!18}{32\!\cdots\!95}a^{20}-\frac{10\!\cdots\!82}{17\!\cdots\!05}a^{19}+\frac{19\!\cdots\!76}{69\!\cdots\!85}a^{18}-\frac{13\!\cdots\!27}{17\!\cdots\!05}a^{17}+\frac{28\!\cdots\!88}{17\!\cdots\!05}a^{16}-\frac{86\!\cdots\!64}{32\!\cdots\!95}a^{15}+\frac{34\!\cdots\!17}{10\!\cdots\!65}a^{14}-\frac{70\!\cdots\!34}{79\!\cdots\!95}a^{13}+\frac{79\!\cdots\!97}{36\!\cdots\!55}a^{12}-\frac{71\!\cdots\!90}{16\!\cdots\!41}a^{11}+\frac{20\!\cdots\!49}{32\!\cdots\!95}a^{10}-\frac{24\!\cdots\!08}{32\!\cdots\!95}a^{9}+\frac{33\!\cdots\!94}{32\!\cdots\!95}a^{8}-\frac{56\!\cdots\!59}{32\!\cdots\!95}a^{7}+\frac{16\!\cdots\!83}{10\!\cdots\!65}a^{6}-\frac{46\!\cdots\!68}{32\!\cdots\!95}a^{5}+\frac{11\!\cdots\!57}{32\!\cdots\!95}a^{4}-\frac{49\!\cdots\!82}{17\!\cdots\!05}a^{3}-\frac{51\!\cdots\!39}{28\!\cdots\!27}a^{2}-\frac{59\!\cdots\!13}{72\!\cdots\!11}a+\frac{84\!\cdots\!58}{24\!\cdots\!37}$, $\frac{15\!\cdots\!38}{34\!\cdots\!95}a^{20}-\frac{36\!\cdots\!74}{12\!\cdots\!35}a^{19}+\frac{72\!\cdots\!86}{50\!\cdots\!95}a^{18}-\frac{53\!\cdots\!52}{12\!\cdots\!35}a^{17}+\frac{12\!\cdots\!94}{12\!\cdots\!35}a^{16}-\frac{39\!\cdots\!39}{23\!\cdots\!65}a^{15}+\frac{10\!\cdots\!71}{46\!\cdots\!15}a^{14}-\frac{40\!\cdots\!81}{83\!\cdots\!95}a^{13}+\frac{32\!\cdots\!38}{26\!\cdots\!85}a^{12}-\frac{31\!\cdots\!57}{12\!\cdots\!07}a^{11}+\frac{95\!\cdots\!87}{23\!\cdots\!65}a^{10}-\frac{12\!\cdots\!16}{23\!\cdots\!65}a^{9}+\frac{16\!\cdots\!71}{23\!\cdots\!65}a^{8}-\frac{25\!\cdots\!44}{23\!\cdots\!65}a^{7}+\frac{13\!\cdots\!31}{11\!\cdots\!65}a^{6}-\frac{51\!\cdots\!52}{47\!\cdots\!73}a^{5}+\frac{14\!\cdots\!61}{23\!\cdots\!65}a^{4}-\frac{29\!\cdots\!74}{12\!\cdots\!35}a^{3}+\frac{12\!\cdots\!23}{20\!\cdots\!29}a^{2}-\frac{18\!\cdots\!05}{52\!\cdots\!97}a+\frac{48\!\cdots\!94}{17\!\cdots\!99}$, $\frac{35\!\cdots\!34}{61\!\cdots\!35}a^{20}-\frac{11\!\cdots\!14}{32\!\cdots\!65}a^{19}+\frac{22\!\cdots\!59}{13\!\cdots\!05}a^{18}-\frac{16\!\cdots\!42}{32\!\cdots\!65}a^{17}+\frac{36\!\cdots\!28}{32\!\cdots\!65}a^{16}-\frac{11\!\cdots\!74}{61\!\cdots\!35}a^{15}+\frac{49\!\cdots\!19}{20\!\cdots\!45}a^{14}-\frac{85\!\cdots\!74}{14\!\cdots\!35}a^{13}+\frac{30\!\cdots\!18}{20\!\cdots\!45}a^{12}-\frac{61\!\cdots\!84}{20\!\cdots\!45}a^{11}+\frac{56\!\cdots\!18}{12\!\cdots\!07}a^{10}-\frac{70\!\cdots\!83}{12\!\cdots\!07}a^{9}+\frac{46\!\cdots\!44}{61\!\cdots\!35}a^{8}-\frac{75\!\cdots\!43}{61\!\cdots\!35}a^{7}+\frac{26\!\cdots\!49}{20\!\cdots\!45}a^{6}-\frac{67\!\cdots\!51}{61\!\cdots\!35}a^{5}+\frac{33\!\cdots\!26}{61\!\cdots\!35}a^{4}-\frac{67\!\cdots\!77}{32\!\cdots\!65}a^{3}+\frac{51\!\cdots\!25}{52\!\cdots\!11}a^{2}-\frac{65\!\cdots\!28}{13\!\cdots\!23}a+\frac{15\!\cdots\!73}{45\!\cdots\!41}$, $\frac{32\!\cdots\!36}{28\!\cdots\!69}a^{20}-\frac{89\!\cdots\!72}{12\!\cdots\!35}a^{19}+\frac{47\!\cdots\!09}{14\!\cdots\!37}a^{18}-\frac{11\!\cdots\!86}{12\!\cdots\!35}a^{17}+\frac{22\!\cdots\!98}{12\!\cdots\!35}a^{16}-\frac{61\!\cdots\!47}{23\!\cdots\!65}a^{15}+\frac{19\!\cdots\!34}{79\!\cdots\!55}a^{14}-\frac{47\!\cdots\!19}{58\!\cdots\!65}a^{13}+\frac{22\!\cdots\!32}{88\!\cdots\!95}a^{12}-\frac{40\!\cdots\!81}{87\!\cdots\!05}a^{11}+\frac{80\!\cdots\!19}{14\!\cdots\!45}a^{10}-\frac{13\!\cdots\!69}{23\!\cdots\!65}a^{9}+\frac{36\!\cdots\!29}{47\!\cdots\!73}a^{8}-\frac{38\!\cdots\!33}{23\!\cdots\!65}a^{7}+\frac{10\!\cdots\!94}{79\!\cdots\!55}a^{6}-\frac{23\!\cdots\!29}{23\!\cdots\!65}a^{5}-\frac{12\!\cdots\!21}{34\!\cdots\!95}a^{4}+\frac{34\!\cdots\!71}{12\!\cdots\!35}a^{3}+\frac{80\!\cdots\!52}{20\!\cdots\!29}a^{2}-\frac{41\!\cdots\!11}{31\!\cdots\!41}a-\frac{35\!\cdots\!46}{17\!\cdots\!99}$, $\frac{92\!\cdots\!91}{23\!\cdots\!65}a^{20}-\frac{24\!\cdots\!14}{12\!\cdots\!35}a^{19}+\frac{43\!\cdots\!26}{50\!\cdots\!95}a^{18}-\frac{24\!\cdots\!69}{12\!\cdots\!35}a^{17}+\frac{40\!\cdots\!67}{12\!\cdots\!35}a^{16}-\frac{86\!\cdots\!84}{23\!\cdots\!65}a^{15}+\frac{84\!\cdots\!19}{79\!\cdots\!55}a^{14}-\frac{12\!\cdots\!94}{58\!\cdots\!65}a^{13}+\frac{32\!\cdots\!79}{58\!\cdots\!33}a^{12}-\frac{93\!\cdots\!16}{12\!\cdots\!07}a^{11}+\frac{15\!\cdots\!48}{23\!\cdots\!65}a^{10}-\frac{16\!\cdots\!03}{47\!\cdots\!73}a^{9}+\frac{19\!\cdots\!28}{23\!\cdots\!65}a^{8}-\frac{66\!\cdots\!93}{23\!\cdots\!65}a^{7}-\frac{61\!\cdots\!11}{79\!\cdots\!55}a^{6}+\frac{54\!\cdots\!52}{23\!\cdots\!65}a^{5}-\frac{61\!\cdots\!24}{23\!\cdots\!65}a^{4}+\frac{11\!\cdots\!71}{73\!\cdots\!55}a^{3}+\frac{14\!\cdots\!67}{20\!\cdots\!29}a^{2}+\frac{24\!\cdots\!25}{52\!\cdots\!97}a-\frac{85\!\cdots\!65}{17\!\cdots\!99}$, $\frac{34\!\cdots\!87}{47\!\cdots\!73}a^{20}-\frac{56\!\cdots\!72}{12\!\cdots\!35}a^{19}+\frac{10\!\cdots\!19}{50\!\cdots\!95}a^{18}-\frac{10\!\cdots\!76}{17\!\cdots\!05}a^{17}+\frac{32\!\cdots\!18}{25\!\cdots\!67}a^{16}-\frac{48\!\cdots\!02}{23\!\cdots\!65}a^{15}+\frac{18\!\cdots\!16}{79\!\cdots\!55}a^{14}-\frac{36\!\cdots\!49}{58\!\cdots\!65}a^{13}+\frac{15\!\cdots\!34}{88\!\cdots\!95}a^{12}-\frac{20\!\cdots\!04}{61\!\cdots\!35}a^{11}+\frac{16\!\cdots\!17}{34\!\cdots\!95}a^{10}-\frac{13\!\cdots\!42}{23\!\cdots\!65}a^{9}+\frac{17\!\cdots\!23}{23\!\cdots\!65}a^{8}-\frac{30\!\cdots\!13}{23\!\cdots\!65}a^{7}+\frac{97\!\cdots\!58}{79\!\cdots\!55}a^{6}-\frac{17\!\cdots\!11}{23\!\cdots\!65}a^{5}+\frac{41\!\cdots\!42}{23\!\cdots\!65}a^{4}+\frac{10\!\cdots\!51}{17\!\cdots\!05}a^{3}+\frac{10\!\cdots\!64}{20\!\cdots\!29}a^{2}-\frac{21\!\cdots\!64}{52\!\cdots\!97}a-\frac{20\!\cdots\!95}{17\!\cdots\!99}$, $\frac{61\!\cdots\!06}{34\!\cdots\!95}a^{20}-\frac{14\!\cdots\!73}{12\!\cdots\!35}a^{19}+\frac{29\!\cdots\!23}{50\!\cdots\!95}a^{18}-\frac{43\!\cdots\!01}{25\!\cdots\!67}a^{17}+\frac{49\!\cdots\!13}{12\!\cdots\!35}a^{16}-\frac{16\!\cdots\!59}{23\!\cdots\!65}a^{15}+\frac{10\!\cdots\!13}{11\!\cdots\!65}a^{14}-\frac{32\!\cdots\!30}{16\!\cdots\!79}a^{13}+\frac{13\!\cdots\!51}{26\!\cdots\!85}a^{12}-\frac{64\!\cdots\!68}{61\!\cdots\!35}a^{11}+\frac{39\!\cdots\!46}{23\!\cdots\!65}a^{10}-\frac{50\!\cdots\!07}{23\!\cdots\!65}a^{9}+\frac{65\!\cdots\!67}{23\!\cdots\!65}a^{8}-\frac{14\!\cdots\!67}{34\!\cdots\!95}a^{7}+\frac{55\!\cdots\!01}{11\!\cdots\!65}a^{6}-\frac{10\!\cdots\!43}{23\!\cdots\!65}a^{5}+\frac{53\!\cdots\!88}{23\!\cdots\!65}a^{4}-\frac{19\!\cdots\!16}{25\!\cdots\!67}a^{3}+\frac{45\!\cdots\!15}{20\!\cdots\!29}a^{2}-\frac{80\!\cdots\!94}{52\!\cdots\!97}a+\frac{23\!\cdots\!47}{25\!\cdots\!57}$, $\frac{43\!\cdots\!17}{18\!\cdots\!05}a^{20}-\frac{17\!\cdots\!22}{96\!\cdots\!95}a^{19}+\frac{50\!\cdots\!62}{55\!\cdots\!45}a^{18}-\frac{29\!\cdots\!78}{96\!\cdots\!95}a^{17}+\frac{14\!\cdots\!14}{19\!\cdots\!59}a^{16}-\frac{37\!\cdots\!22}{26\!\cdots\!15}a^{15}+\frac{12\!\cdots\!04}{61\!\cdots\!35}a^{14}-\frac{17\!\cdots\!16}{44\!\cdots\!05}a^{13}+\frac{19\!\cdots\!41}{20\!\cdots\!45}a^{12}-\frac{18\!\cdots\!03}{87\!\cdots\!05}a^{11}+\frac{66\!\cdots\!39}{18\!\cdots\!05}a^{10}-\frac{94\!\cdots\!61}{18\!\cdots\!05}a^{9}+\frac{35\!\cdots\!01}{52\!\cdots\!03}a^{8}-\frac{17\!\cdots\!56}{18\!\cdots\!05}a^{7}+\frac{76\!\cdots\!78}{61\!\cdots\!35}a^{6}-\frac{23\!\cdots\!97}{18\!\cdots\!05}a^{5}+\frac{24\!\cdots\!76}{26\!\cdots\!15}a^{4}-\frac{55\!\cdots\!70}{11\!\cdots\!27}a^{3}+\frac{26\!\cdots\!82}{15\!\cdots\!33}a^{2}-\frac{28\!\cdots\!39}{58\!\cdots\!67}a+\frac{14\!\cdots\!32}{13\!\cdots\!23}$, $\frac{89\!\cdots\!21}{11\!\cdots\!65}a^{20}-\frac{22\!\cdots\!94}{41\!\cdots\!45}a^{19}+\frac{93\!\cdots\!32}{33\!\cdots\!53}a^{18}-\frac{37\!\cdots\!54}{41\!\cdots\!45}a^{17}+\frac{18\!\cdots\!72}{83\!\cdots\!89}a^{16}-\frac{32\!\cdots\!53}{79\!\cdots\!55}a^{15}+\frac{15\!\cdots\!57}{26\!\cdots\!85}a^{14}-\frac{31\!\cdots\!64}{27\!\cdots\!65}a^{13}+\frac{72\!\cdots\!03}{26\!\cdots\!85}a^{12}-\frac{12\!\cdots\!86}{20\!\cdots\!45}a^{11}+\frac{80\!\cdots\!13}{79\!\cdots\!55}a^{10}-\frac{11\!\cdots\!04}{79\!\cdots\!55}a^{9}+\frac{15\!\cdots\!36}{79\!\cdots\!55}a^{8}-\frac{44\!\cdots\!38}{15\!\cdots\!91}a^{7}+\frac{13\!\cdots\!72}{37\!\cdots\!55}a^{6}-\frac{28\!\cdots\!03}{79\!\cdots\!55}a^{5}+\frac{42\!\cdots\!83}{15\!\cdots\!91}a^{4}-\frac{66\!\cdots\!46}{41\!\cdots\!45}a^{3}+\frac{45\!\cdots\!80}{68\!\cdots\!43}a^{2}-\frac{41\!\cdots\!13}{17\!\cdots\!99}a+\frac{37\!\cdots\!86}{58\!\cdots\!33}$, $\frac{86\!\cdots\!47}{47\!\cdots\!73}a^{20}-\frac{15\!\cdots\!13}{12\!\cdots\!35}a^{19}+\frac{43\!\cdots\!62}{72\!\cdots\!85}a^{18}-\frac{22\!\cdots\!06}{12\!\cdots\!35}a^{17}+\frac{10\!\cdots\!97}{25\!\cdots\!67}a^{16}-\frac{16\!\cdots\!69}{23\!\cdots\!65}a^{15}+\frac{14\!\cdots\!33}{15\!\cdots\!91}a^{14}-\frac{11\!\cdots\!82}{58\!\cdots\!65}a^{13}+\frac{51\!\cdots\!07}{98\!\cdots\!55}a^{12}-\frac{95\!\cdots\!78}{87\!\cdots\!05}a^{11}+\frac{39\!\cdots\!58}{23\!\cdots\!65}a^{10}-\frac{49\!\cdots\!83}{23\!\cdots\!65}a^{9}+\frac{63\!\cdots\!29}{23\!\cdots\!65}a^{8}-\frac{10\!\cdots\!51}{23\!\cdots\!65}a^{7}+\frac{40\!\cdots\!66}{79\!\cdots\!55}a^{6}-\frac{88\!\cdots\!23}{23\!\cdots\!65}a^{5}+\frac{61\!\cdots\!11}{34\!\cdots\!95}a^{4}-\frac{66\!\cdots\!34}{12\!\cdots\!35}a^{3}+\frac{34\!\cdots\!95}{20\!\cdots\!29}a^{2}-\frac{62\!\cdots\!30}{52\!\cdots\!97}a+\frac{42\!\cdots\!96}{17\!\cdots\!99}$
| 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: | \( 859506502.583 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 859506502.583 \cdot 1}{2\cdot\sqrt{2807535013090453863383795166015625}}\cr\approx \mathstrut & 1.55555460091 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405)
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 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405, 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 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405);
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 - 7*x^20 + 35*x^19 - 112*x^18 + 266*x^17 - 490*x^16 + 693*x^15 - 1355*x^14 + 3360*x^13 - 7224*x^12 + 12166*x^11 - 16709*x^10 + 21700*x^9 - 32417*x^8 + 39987*x^7 - 39067*x^6 + 26600*x^5 - 12320*x^4 + 4389*x^3 - 1575*x^2 + 945*x - 405);
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
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 42 |
| The 12 conjugacy class representatives for $D_{21}$ |
| Character table for $D_{21}$ |
Intermediate fields
| 3.1.175.1, 7.1.40353607.1 |
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
| Galois closure: | deg 42 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | ${\href{/padicField/3.2.0.1}{2} }^{10}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | R | $21$ | ${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{10}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/31.2.0.1}{2} }^{10}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21$ | ${\href{/padicField/47.2.0.1}{2} }^{10}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(7\)
| 7.7.9.4 | $x^{7} + 35 x^{3} + 7$ | $7$ | $1$ | $9$ | $D_{7}$ | $[3/2]_{2}$ |
| 7.14.19.18 | $x^{14} + 35 x^{6} + 7$ | $14$ | $1$ | $19$ | $D_{7}$ | $[3/2]_{2}$ |