Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447)
gp: K = bnfinit(y^21 - 7*y^20 + 28*y^19 - 140*y^18 + 539*y^17 - 1589*y^16 + 5208*y^15 - 14550*y^14 + 33726*y^13 - 85652*y^12 + 199136*y^11 - 392630*y^10 + 782026*y^9 - 779296*y^8 + 517684*y^7 - 1556478*y^6 + 6082125*y^5 - 9280761*y^4 + 10378536*y^3 - 3684954*y^2 + 2233119*y - 4362447, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447)
\( x^{21} - 7 x^{20} + 28 x^{19} - 140 x^{18} + 539 x^{17} - 1589 x^{16} + 5208 x^{15} - 14550 x^{14} + \cdots - 4362447 \)
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: |
\(11841910006204867301303769538659846667225792512\)
\(\medspace = 2^{26}\cdot 3^{19}\cdot 7^{21}\cdot 43^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(156.30\) | 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: | $2^{10/7}3^{13/14}7^{47/42}43^{1/2}\approx 432.03923993641814$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{903}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{13}+\frac{1}{6}a^{11}-\frac{1}{12}a^{10}-\frac{1}{4}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{15}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}-\frac{1}{4}a^{11}-\frac{1}{6}a^{10}+\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{12}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{24}a^{16}-\frac{1}{4}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{8}$, $\frac{1}{24}a^{17}-\frac{1}{4}a^{9}+\frac{1}{6}a^{8}-\frac{1}{2}a^{3}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{24}a^{18}-\frac{1}{4}a^{10}+\frac{1}{6}a^{9}-\frac{1}{2}a^{4}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{72}a^{19}+\frac{1}{72}a^{16}+\frac{1}{12}a^{13}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{9}a^{10}-\frac{1}{12}a^{9}+\frac{1}{6}a^{8}+\frac{1}{18}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{24}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{5}{24}$, $\frac{1}{15\!\cdots\!52}a^{20}+\frac{22\!\cdots\!19}{38\!\cdots\!38}a^{19}+\frac{39\!\cdots\!95}{50\!\cdots\!84}a^{18}-\frac{10\!\cdots\!03}{76\!\cdots\!76}a^{17}+\frac{23\!\cdots\!81}{15\!\cdots\!52}a^{16}+\frac{50\!\cdots\!95}{25\!\cdots\!92}a^{15}+\frac{10\!\cdots\!37}{63\!\cdots\!23}a^{14}+\frac{89\!\cdots\!65}{25\!\cdots\!92}a^{13}-\frac{13\!\cdots\!25}{12\!\cdots\!46}a^{12}+\frac{16\!\cdots\!45}{76\!\cdots\!76}a^{11}+\frac{15\!\cdots\!03}{76\!\cdots\!76}a^{10}+\frac{10\!\cdots\!51}{84\!\cdots\!64}a^{9}-\frac{20\!\cdots\!98}{19\!\cdots\!69}a^{8}+\frac{53\!\cdots\!31}{76\!\cdots\!76}a^{7}+\frac{18\!\cdots\!05}{42\!\cdots\!82}a^{6}+\frac{23\!\cdots\!55}{84\!\cdots\!64}a^{5}+\frac{22\!\cdots\!57}{50\!\cdots\!84}a^{4}-\frac{12\!\cdots\!05}{25\!\cdots\!92}a^{3}-\frac{39\!\cdots\!43}{16\!\cdots\!28}a^{2}-\frac{93\!\cdots\!09}{12\!\cdots\!46}a+\frac{14\!\cdots\!57}{50\!\cdots\!84}$
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
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: |
$\frac{42\!\cdots\!87}{42\!\cdots\!82}a^{20}-\frac{38\!\cdots\!48}{63\!\cdots\!23}a^{19}+\frac{14\!\cdots\!81}{63\!\cdots\!23}a^{18}-\frac{30\!\cdots\!09}{25\!\cdots\!92}a^{17}+\frac{36\!\cdots\!77}{84\!\cdots\!64}a^{16}-\frac{75\!\cdots\!16}{63\!\cdots\!23}a^{15}+\frac{26\!\cdots\!62}{63\!\cdots\!23}a^{14}-\frac{68\!\cdots\!43}{63\!\cdots\!23}a^{13}+\frac{15\!\cdots\!38}{63\!\cdots\!23}a^{12}-\frac{40\!\cdots\!48}{63\!\cdots\!23}a^{11}+\frac{59\!\cdots\!23}{42\!\cdots\!82}a^{10}-\frac{11\!\cdots\!67}{42\!\cdots\!82}a^{9}+\frac{23\!\cdots\!53}{42\!\cdots\!82}a^{8}-\frac{19\!\cdots\!44}{63\!\cdots\!23}a^{7}+\frac{68\!\cdots\!13}{21\!\cdots\!41}a^{6}-\frac{28\!\cdots\!71}{21\!\cdots\!41}a^{5}+\frac{20\!\cdots\!01}{42\!\cdots\!82}a^{4}-\frac{10\!\cdots\!21}{21\!\cdots\!41}a^{3}+\frac{27\!\cdots\!11}{42\!\cdots\!82}a^{2}+\frac{11\!\cdots\!13}{84\!\cdots\!64}a+\frac{42\!\cdots\!89}{84\!\cdots\!64}$, $\frac{85\!\cdots\!83}{84\!\cdots\!64}a^{20}-\frac{83\!\cdots\!32}{63\!\cdots\!23}a^{19}+\frac{31\!\cdots\!61}{42\!\cdots\!82}a^{18}-\frac{28\!\cdots\!35}{84\!\cdots\!64}a^{17}+\frac{37\!\cdots\!17}{25\!\cdots\!92}a^{16}-\frac{33\!\cdots\!57}{63\!\cdots\!23}a^{15}+\frac{10\!\cdots\!29}{63\!\cdots\!23}a^{14}-\frac{32\!\cdots\!04}{63\!\cdots\!23}a^{13}+\frac{17\!\cdots\!21}{12\!\cdots\!46}a^{12}-\frac{21\!\cdots\!99}{63\!\cdots\!23}a^{11}+\frac{34\!\cdots\!55}{42\!\cdots\!82}a^{10}-\frac{23\!\cdots\!23}{12\!\cdots\!46}a^{9}+\frac{23\!\cdots\!55}{63\!\cdots\!23}a^{8}-\frac{42\!\cdots\!63}{63\!\cdots\!23}a^{7}+\frac{15\!\cdots\!94}{21\!\cdots\!41}a^{6}-\frac{13\!\cdots\!56}{21\!\cdots\!41}a^{5}+\frac{12\!\cdots\!07}{84\!\cdots\!64}a^{4}-\frac{10\!\cdots\!56}{21\!\cdots\!41}a^{3}+\frac{18\!\cdots\!22}{21\!\cdots\!41}a^{2}-\frac{80\!\cdots\!31}{84\!\cdots\!64}a+\frac{38\!\cdots\!41}{84\!\cdots\!64}$, $\frac{30\!\cdots\!57}{84\!\cdots\!64}a^{20}-\frac{26\!\cdots\!77}{12\!\cdots\!46}a^{19}+\frac{47\!\cdots\!40}{63\!\cdots\!23}a^{18}-\frac{34\!\cdots\!55}{84\!\cdots\!64}a^{17}+\frac{36\!\cdots\!01}{25\!\cdots\!92}a^{16}-\frac{25\!\cdots\!32}{63\!\cdots\!23}a^{15}+\frac{88\!\cdots\!08}{63\!\cdots\!23}a^{14}-\frac{22\!\cdots\!89}{63\!\cdots\!23}a^{13}+\frac{49\!\cdots\!49}{63\!\cdots\!23}a^{12}-\frac{13\!\cdots\!73}{63\!\cdots\!23}a^{11}+\frac{19\!\cdots\!49}{42\!\cdots\!82}a^{10}-\frac{11\!\cdots\!65}{12\!\cdots\!46}a^{9}+\frac{11\!\cdots\!67}{63\!\cdots\!23}a^{8}-\frac{49\!\cdots\!84}{63\!\cdots\!23}a^{7}+\frac{28\!\cdots\!11}{21\!\cdots\!41}a^{6}-\frac{99\!\cdots\!47}{21\!\cdots\!41}a^{5}+\frac{15\!\cdots\!59}{84\!\cdots\!64}a^{4}-\frac{61\!\cdots\!21}{42\!\cdots\!82}a^{3}+\frac{87\!\cdots\!87}{42\!\cdots\!82}a^{2}+\frac{82\!\cdots\!57}{84\!\cdots\!64}a+\frac{14\!\cdots\!53}{84\!\cdots\!64}$, $\frac{81\!\cdots\!71}{12\!\cdots\!46}a^{20}-\frac{16\!\cdots\!56}{63\!\cdots\!23}a^{19}+\frac{12\!\cdots\!29}{21\!\cdots\!41}a^{18}-\frac{93\!\cdots\!28}{21\!\cdots\!41}a^{17}+\frac{13\!\cdots\!09}{12\!\cdots\!46}a^{16}-\frac{86\!\cdots\!33}{63\!\cdots\!23}a^{15}+\frac{55\!\cdots\!29}{63\!\cdots\!23}a^{14}-\frac{61\!\cdots\!37}{63\!\cdots\!23}a^{13}-\frac{70\!\cdots\!50}{63\!\cdots\!23}a^{12}-\frac{10\!\cdots\!35}{21\!\cdots\!41}a^{11}-\frac{34\!\cdots\!29}{63\!\cdots\!23}a^{10}+\frac{33\!\cdots\!67}{63\!\cdots\!23}a^{9}-\frac{79\!\cdots\!37}{12\!\cdots\!46}a^{8}+\frac{42\!\cdots\!17}{63\!\cdots\!23}a^{7}-\frac{11\!\cdots\!55}{21\!\cdots\!41}a^{6}-\frac{10\!\cdots\!17}{21\!\cdots\!41}a^{5}+\frac{14\!\cdots\!41}{42\!\cdots\!82}a^{4}+\frac{89\!\cdots\!27}{21\!\cdots\!41}a^{3}-\frac{99\!\cdots\!66}{21\!\cdots\!41}a^{2}+\frac{37\!\cdots\!66}{21\!\cdots\!41}a-\frac{35\!\cdots\!29}{21\!\cdots\!41}$, $\frac{44\!\cdots\!07}{16\!\cdots\!28}a^{20}-\frac{90\!\cdots\!69}{50\!\cdots\!84}a^{19}+\frac{34\!\cdots\!21}{50\!\cdots\!84}a^{18}-\frac{44\!\cdots\!61}{12\!\cdots\!46}a^{17}+\frac{67\!\cdots\!93}{50\!\cdots\!84}a^{16}-\frac{31\!\cdots\!89}{84\!\cdots\!64}a^{15}+\frac{31\!\cdots\!93}{25\!\cdots\!92}a^{14}-\frac{87\!\cdots\!63}{25\!\cdots\!92}a^{13}+\frac{15\!\cdots\!55}{21\!\cdots\!41}a^{12}-\frac{24\!\cdots\!47}{12\!\cdots\!46}a^{11}+\frac{28\!\cdots\!15}{63\!\cdots\!23}a^{10}-\frac{20\!\cdots\!93}{25\!\cdots\!92}a^{9}+\frac{10\!\cdots\!55}{63\!\cdots\!23}a^{8}-\frac{33\!\cdots\!47}{25\!\cdots\!92}a^{7}-\frac{16\!\cdots\!23}{84\!\cdots\!64}a^{6}-\frac{17\!\cdots\!83}{84\!\cdots\!64}a^{5}+\frac{23\!\cdots\!29}{16\!\cdots\!28}a^{4}-\frac{36\!\cdots\!09}{16\!\cdots\!28}a^{3}+\frac{30\!\cdots\!89}{16\!\cdots\!28}a^{2}+\frac{74\!\cdots\!47}{84\!\cdots\!64}a-\frac{28\!\cdots\!59}{16\!\cdots\!28}$, $\frac{93\!\cdots\!27}{15\!\cdots\!52}a^{20}-\frac{56\!\cdots\!17}{76\!\cdots\!76}a^{19}+\frac{16\!\cdots\!51}{50\!\cdots\!84}a^{18}-\frac{25\!\cdots\!74}{19\!\cdots\!69}a^{17}+\frac{23\!\cdots\!35}{38\!\cdots\!38}a^{16}-\frac{47\!\cdots\!49}{25\!\cdots\!92}a^{15}+\frac{22\!\cdots\!73}{42\!\cdots\!82}a^{14}-\frac{43\!\cdots\!67}{25\!\cdots\!92}a^{13}+\frac{50\!\cdots\!85}{12\!\cdots\!46}a^{12}-\frac{64\!\cdots\!31}{76\!\cdots\!76}a^{11}+\frac{17\!\cdots\!07}{76\!\cdots\!76}a^{10}-\frac{39\!\cdots\!53}{84\!\cdots\!64}a^{9}+\frac{57\!\cdots\!13}{76\!\cdots\!76}a^{8}-\frac{94\!\cdots\!13}{76\!\cdots\!76}a^{7}-\frac{11\!\cdots\!06}{21\!\cdots\!41}a^{6}-\frac{43\!\cdots\!81}{84\!\cdots\!64}a^{5}+\frac{49\!\cdots\!47}{50\!\cdots\!84}a^{4}-\frac{96\!\cdots\!28}{63\!\cdots\!23}a^{3}-\frac{49\!\cdots\!67}{16\!\cdots\!28}a^{2}-\frac{14\!\cdots\!95}{25\!\cdots\!92}a+\frac{10\!\cdots\!37}{25\!\cdots\!92}$, $\frac{23\!\cdots\!29}{50\!\cdots\!84}a^{20}-\frac{16\!\cdots\!35}{50\!\cdots\!84}a^{19}+\frac{21\!\cdots\!59}{16\!\cdots\!28}a^{18}-\frac{32\!\cdots\!91}{50\!\cdots\!84}a^{17}+\frac{10\!\cdots\!17}{42\!\cdots\!82}a^{16}-\frac{45\!\cdots\!66}{63\!\cdots\!23}a^{15}+\frac{14\!\cdots\!93}{63\!\cdots\!23}a^{14}-\frac{56\!\cdots\!71}{84\!\cdots\!64}a^{13}+\frac{38\!\cdots\!77}{25\!\cdots\!92}a^{12}-\frac{98\!\cdots\!93}{25\!\cdots\!92}a^{11}+\frac{23\!\cdots\!33}{25\!\cdots\!92}a^{10}-\frac{11\!\cdots\!17}{63\!\cdots\!23}a^{9}+\frac{23\!\cdots\!46}{63\!\cdots\!23}a^{8}-\frac{16\!\cdots\!29}{42\!\cdots\!82}a^{7}+\frac{12\!\cdots\!67}{42\!\cdots\!82}a^{6}-\frac{86\!\cdots\!11}{84\!\cdots\!64}a^{5}+\frac{62\!\cdots\!31}{16\!\cdots\!28}a^{4}-\frac{92\!\cdots\!61}{16\!\cdots\!28}a^{3}+\frac{79\!\cdots\!31}{16\!\cdots\!28}a^{2}+\frac{41\!\cdots\!53}{16\!\cdots\!28}a-\frac{47\!\cdots\!54}{21\!\cdots\!41}$, $\frac{17\!\cdots\!44}{63\!\cdots\!23}a^{20}-\frac{92\!\cdots\!45}{76\!\cdots\!76}a^{19}+\frac{80\!\cdots\!36}{21\!\cdots\!41}a^{18}-\frac{55\!\cdots\!59}{21\!\cdots\!41}a^{17}+\frac{29\!\cdots\!59}{38\!\cdots\!38}a^{16}-\frac{46\!\cdots\!39}{25\!\cdots\!92}a^{15}+\frac{20\!\cdots\!23}{25\!\cdots\!92}a^{14}-\frac{44\!\cdots\!35}{25\!\cdots\!92}a^{13}+\frac{26\!\cdots\!37}{84\!\cdots\!64}a^{12}-\frac{30\!\cdots\!55}{25\!\cdots\!92}a^{11}+\frac{17\!\cdots\!45}{76\!\cdots\!76}a^{10}-\frac{68\!\cdots\!57}{25\!\cdots\!92}a^{9}+\frac{23\!\cdots\!79}{25\!\cdots\!92}a^{8}-\frac{41\!\cdots\!01}{76\!\cdots\!76}a^{7}+\frac{43\!\cdots\!25}{84\!\cdots\!64}a^{6}-\frac{85\!\cdots\!01}{84\!\cdots\!64}a^{5}+\frac{97\!\cdots\!77}{84\!\cdots\!64}a^{4}-\frac{30\!\cdots\!91}{63\!\cdots\!23}a^{3}+\frac{10\!\cdots\!09}{84\!\cdots\!64}a^{2}+\frac{16\!\cdots\!89}{84\!\cdots\!64}a+\frac{47\!\cdots\!29}{25\!\cdots\!92}$, $\frac{54\!\cdots\!53}{15\!\cdots\!52}a^{20}-\frac{35\!\cdots\!14}{19\!\cdots\!69}a^{19}+\frac{82\!\cdots\!35}{16\!\cdots\!28}a^{18}-\frac{41\!\cdots\!47}{15\!\cdots\!52}a^{17}+\frac{12\!\cdots\!91}{15\!\cdots\!52}a^{16}-\frac{16\!\cdots\!53}{12\!\cdots\!46}a^{15}+\frac{31\!\cdots\!86}{63\!\cdots\!23}a^{14}-\frac{21\!\cdots\!03}{25\!\cdots\!92}a^{13}-\frac{14\!\cdots\!97}{25\!\cdots\!92}a^{12}+\frac{59\!\cdots\!70}{19\!\cdots\!69}a^{11}-\frac{39\!\cdots\!61}{76\!\cdots\!76}a^{10}+\frac{85\!\cdots\!47}{21\!\cdots\!41}a^{9}-\frac{73\!\cdots\!55}{76\!\cdots\!76}a^{8}+\frac{17\!\cdots\!77}{38\!\cdots\!38}a^{7}-\frac{16\!\cdots\!47}{21\!\cdots\!41}a^{6}+\frac{19\!\cdots\!43}{84\!\cdots\!64}a^{5}+\frac{51\!\cdots\!63}{50\!\cdots\!84}a^{4}+\frac{15\!\cdots\!33}{12\!\cdots\!46}a^{3}-\frac{10\!\cdots\!97}{16\!\cdots\!28}a^{2}+\frac{58\!\cdots\!05}{50\!\cdots\!84}a-\frac{38\!\cdots\!91}{50\!\cdots\!84}$, $\frac{13\!\cdots\!71}{15\!\cdots\!52}a^{20}-\frac{10\!\cdots\!98}{19\!\cdots\!69}a^{19}+\frac{96\!\cdots\!59}{50\!\cdots\!84}a^{18}-\frac{15\!\cdots\!67}{15\!\cdots\!52}a^{17}+\frac{56\!\cdots\!69}{15\!\cdots\!52}a^{16}-\frac{25\!\cdots\!81}{25\!\cdots\!92}a^{15}+\frac{29\!\cdots\!85}{84\!\cdots\!64}a^{14}-\frac{23\!\cdots\!79}{25\!\cdots\!92}a^{13}+\frac{40\!\cdots\!22}{21\!\cdots\!41}a^{12}-\frac{41\!\cdots\!33}{76\!\cdots\!76}a^{11}+\frac{44\!\cdots\!89}{38\!\cdots\!38}a^{10}-\frac{26\!\cdots\!31}{12\!\cdots\!46}a^{9}+\frac{17\!\cdots\!31}{38\!\cdots\!38}a^{8}-\frac{12\!\cdots\!17}{76\!\cdots\!76}a^{7}+\frac{53\!\cdots\!81}{84\!\cdots\!64}a^{6}-\frac{14\!\cdots\!91}{84\!\cdots\!64}a^{5}+\frac{16\!\cdots\!35}{50\!\cdots\!84}a^{4}-\frac{12\!\cdots\!27}{25\!\cdots\!92}a^{3}+\frac{45\!\cdots\!79}{16\!\cdots\!28}a^{2}+\frac{18\!\cdots\!93}{50\!\cdots\!84}a+\frac{14\!\cdots\!89}{50\!\cdots\!84}$
| 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: | \( 2613750744265613.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 2613750744265613.5 \cdot 1}{2\cdot\sqrt{11841910006204867301303769538659846667225792512}}\cr\approx \mathstrut & 2.30330778206600 \end{aligned}\] (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 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447)
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 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447, 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 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447);
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 + 28*x^19 - 140*x^18 + 539*x^17 - 1589*x^16 + 5208*x^15 - 14550*x^14 + 33726*x^13 - 85652*x^12 + 199136*x^11 - 392630*x^10 + 782026*x^9 - 779296*x^8 + 517684*x^7 - 1556478*x^6 + 6082125*x^5 - 9280761*x^4 + 10378536*x^3 - 3684954*x^2 + 2233119*x - 4362447);
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
$S_3\times F_7$ (as 21T15):
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 252 |
| The 21 conjugacy class representatives for $S_3\times F_7$ |
| Character table for $S_3\times F_7$ |
Intermediate fields
| 3.1.516.1, 7.1.38423222208.8 |
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 42 siblings: | data not computed |
| 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | R | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.14.20.8 | $x^{14} + 2 x^{13} + 2 x^{10} + 2 x^{7} + 2$ | $14$ | $1$ | $20$ | $(C_7:C_3) \times C_2$ | $[2]_{7}^{3}$ | |
|
\(3\)
| 3.7.6.1 | $x^{7} + 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ |
| 3.14.13.2 | $x^{14} + 3$ | $14$ | $1$ | $13$ | $F_7 \times C_2$ | $[\ ]_{14}^{6}$ | |
|
\(7\)
| Deg $21$ | $7$ | $3$ | $21$ | |||
|
\(43\)
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |