Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1)
gp: K = bnfinit(y^21 - 7*y^20 - 13*y^19 + 208*y^18 - 267*y^17 - 1890*y^16 + 5310*y^15 + 4781*y^14 - 31996*y^13 + 16294*y^12 + 77939*y^11 - 103109*y^10 - 52599*y^9 + 164304*y^8 - 52267*y^7 - 74030*y^6 + 58226*y^5 - 9947*y^4 - 1399*y^3 + 399*y^2 - 10*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 7*x^20 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1)
\( x^{21} - 7 x^{20} - 13 x^{19} + 208 x^{18} - 267 x^{17} - 1890 x^{16} + 5310 x^{15} + 4781 x^{14} + \cdots - 1 \)
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: |
\(31993368160435329741277209296896\)
\(\medspace = 2^{18}\cdot 73^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.64\) | 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: | $2^{6/7}73^{2/3}\approx 31.640326082417378$ | ||
| Ramified primes: |
\(2\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $21$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{16\!\cdots\!65}a^{20}+\frac{58123111048383}{16\!\cdots\!65}a^{19}+\frac{3734999696176}{16\!\cdots\!65}a^{18}-\frac{515714038752107}{16\!\cdots\!65}a^{17}+\frac{375135922018768}{16\!\cdots\!65}a^{16}+\frac{116618451789255}{323617895320073}a^{15}+\frac{376267587928901}{16\!\cdots\!65}a^{14}-\frac{140697310776285}{323617895320073}a^{13}-\frac{766626976463449}{16\!\cdots\!65}a^{12}-\frac{753501798021011}{16\!\cdots\!65}a^{11}-\frac{286446107718676}{16\!\cdots\!65}a^{10}+\frac{130652890560010}{323617895320073}a^{9}-\frac{224750701894986}{16\!\cdots\!65}a^{8}-\frac{602348861250697}{16\!\cdots\!65}a^{7}+\frac{226197888251838}{16\!\cdots\!65}a^{6}-\frac{202470394756069}{16\!\cdots\!65}a^{5}+\frac{208504001442222}{16\!\cdots\!65}a^{4}-\frac{122526706673186}{16\!\cdots\!65}a^{3}-\frac{110091881694321}{323617895320073}a^{2}-\frac{514522742215396}{16\!\cdots\!65}a-\frac{485200243909626}{16\!\cdots\!65}$
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: | $20$ | 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: |
$\frac{36\!\cdots\!37}{16\!\cdots\!65}a^{20}-\frac{17\!\cdots\!78}{16\!\cdots\!65}a^{19}-\frac{17\!\cdots\!26}{323617895320073}a^{18}+\frac{11\!\cdots\!67}{323617895320073}a^{17}+\frac{27\!\cdots\!93}{16\!\cdots\!65}a^{16}-\frac{12\!\cdots\!59}{323617895320073}a^{15}+\frac{57\!\cdots\!18}{16\!\cdots\!65}a^{14}+\frac{30\!\cdots\!01}{16\!\cdots\!65}a^{13}-\frac{10\!\cdots\!94}{323617895320073}a^{12}-\frac{55\!\cdots\!03}{16\!\cdots\!65}a^{11}+\frac{17\!\cdots\!71}{16\!\cdots\!65}a^{10}-\frac{10\!\cdots\!94}{16\!\cdots\!65}a^{9}-\frac{22\!\cdots\!93}{16\!\cdots\!65}a^{8}+\frac{12\!\cdots\!63}{16\!\cdots\!65}a^{7}+\frac{86\!\cdots\!06}{16\!\cdots\!65}a^{6}-\frac{90\!\cdots\!54}{16\!\cdots\!65}a^{5}+\frac{33\!\cdots\!68}{323617895320073}a^{4}+\frac{22\!\cdots\!32}{16\!\cdots\!65}a^{3}-\frac{63\!\cdots\!63}{16\!\cdots\!65}a^{2}+\frac{27\!\cdots\!87}{323617895320073}a+\frac{15\!\cdots\!53}{16\!\cdots\!65}$, $\frac{28\!\cdots\!37}{16\!\cdots\!65}a^{20}-\frac{13\!\cdots\!24}{16\!\cdots\!65}a^{19}-\frac{66\!\cdots\!83}{16\!\cdots\!65}a^{18}+\frac{44\!\cdots\!26}{16\!\cdots\!65}a^{17}+\frac{20\!\cdots\!96}{16\!\cdots\!65}a^{16}-\frac{98\!\cdots\!94}{323617895320073}a^{15}+\frac{44\!\cdots\!62}{16\!\cdots\!65}a^{14}+\frac{46\!\cdots\!06}{323617895320073}a^{13}-\frac{40\!\cdots\!68}{16\!\cdots\!65}a^{12}-\frac{41\!\cdots\!67}{16\!\cdots\!65}a^{11}+\frac{13\!\cdots\!88}{16\!\cdots\!65}a^{10}-\frac{20\!\cdots\!22}{323617895320073}a^{9}-\frac{17\!\cdots\!97}{16\!\cdots\!65}a^{8}+\frac{99\!\cdots\!36}{16\!\cdots\!65}a^{7}+\frac{64\!\cdots\!46}{16\!\cdots\!65}a^{6}-\frac{70\!\cdots\!53}{16\!\cdots\!65}a^{5}+\frac{14\!\cdots\!99}{16\!\cdots\!65}a^{4}+\frac{15\!\cdots\!33}{16\!\cdots\!65}a^{3}-\frac{10\!\cdots\!12}{323617895320073}a^{2}+\frac{17\!\cdots\!38}{16\!\cdots\!65}a+\frac{10\!\cdots\!33}{16\!\cdots\!65}$, $\frac{80\!\cdots\!12}{323617895320073}a^{20}-\frac{19\!\cdots\!99}{16\!\cdots\!65}a^{19}-\frac{94\!\cdots\!94}{16\!\cdots\!65}a^{18}+\frac{63\!\cdots\!23}{16\!\cdots\!65}a^{17}+\frac{28\!\cdots\!64}{16\!\cdots\!65}a^{16}-\frac{69\!\cdots\!14}{16\!\cdots\!65}a^{15}+\frac{63\!\cdots\!26}{16\!\cdots\!65}a^{14}+\frac{32\!\cdots\!89}{16\!\cdots\!65}a^{13}-\frac{58\!\cdots\!69}{16\!\cdots\!65}a^{12}-\frac{59\!\cdots\!57}{16\!\cdots\!65}a^{11}+\frac{37\!\cdots\!63}{323617895320073}a^{10}-\frac{26\!\cdots\!78}{323617895320073}a^{9}-\frac{24\!\cdots\!88}{16\!\cdots\!65}a^{8}+\frac{13\!\cdots\!69}{16\!\cdots\!65}a^{7}+\frac{93\!\cdots\!61}{16\!\cdots\!65}a^{6}-\frac{99\!\cdots\!91}{16\!\cdots\!65}a^{5}+\frac{19\!\cdots\!73}{16\!\cdots\!65}a^{4}+\frac{23\!\cdots\!87}{16\!\cdots\!65}a^{3}-\frac{71\!\cdots\!46}{16\!\cdots\!65}a^{2}+\frac{18\!\cdots\!38}{16\!\cdots\!65}a+\frac{18\!\cdots\!36}{16\!\cdots\!65}$, $\frac{36\!\cdots\!72}{323617895320073}a^{20}-\frac{88\!\cdots\!76}{16\!\cdots\!65}a^{19}-\frac{41\!\cdots\!31}{16\!\cdots\!65}a^{18}+\frac{28\!\cdots\!32}{16\!\cdots\!65}a^{17}+\frac{11\!\cdots\!76}{16\!\cdots\!65}a^{16}-\frac{31\!\cdots\!81}{16\!\cdots\!65}a^{15}+\frac{30\!\cdots\!64}{16\!\cdots\!65}a^{14}+\frac{14\!\cdots\!91}{16\!\cdots\!65}a^{13}-\frac{26\!\cdots\!51}{16\!\cdots\!65}a^{12}-\frac{25\!\cdots\!83}{16\!\cdots\!65}a^{11}+\frac{17\!\cdots\!22}{323617895320073}a^{10}-\frac{20\!\cdots\!75}{323617895320073}a^{9}-\frac{10\!\cdots\!82}{16\!\cdots\!65}a^{8}+\frac{68\!\cdots\!06}{16\!\cdots\!65}a^{7}+\frac{39\!\cdots\!34}{16\!\cdots\!65}a^{6}-\frac{47\!\cdots\!04}{16\!\cdots\!65}a^{5}+\frac{10\!\cdots\!67}{16\!\cdots\!65}a^{4}+\frac{77\!\cdots\!48}{16\!\cdots\!65}a^{3}-\frac{39\!\cdots\!39}{16\!\cdots\!65}a^{2}+\frac{17\!\cdots\!57}{16\!\cdots\!65}a+\frac{99\!\cdots\!79}{16\!\cdots\!65}$, $\frac{51\!\cdots\!13}{16\!\cdots\!65}a^{20}-\frac{23\!\cdots\!91}{16\!\cdots\!65}a^{19}-\frac{12\!\cdots\!03}{16\!\cdots\!65}a^{18}+\frac{78\!\cdots\!16}{16\!\cdots\!65}a^{17}+\frac{10\!\cdots\!46}{323617895320073}a^{16}-\frac{88\!\cdots\!37}{16\!\cdots\!65}a^{15}+\frac{61\!\cdots\!43}{16\!\cdots\!65}a^{14}+\frac{43\!\cdots\!29}{16\!\cdots\!65}a^{13}-\frac{65\!\cdots\!68}{16\!\cdots\!65}a^{12}-\frac{92\!\cdots\!06}{16\!\cdots\!65}a^{11}+\frac{22\!\cdots\!83}{16\!\cdots\!65}a^{10}+\frac{35\!\cdots\!67}{16\!\cdots\!65}a^{9}-\frac{31\!\cdots\!84}{16\!\cdots\!65}a^{8}+\frac{21\!\cdots\!69}{323617895320073}a^{7}+\frac{15\!\cdots\!87}{16\!\cdots\!65}a^{6}-\frac{98\!\cdots\!77}{16\!\cdots\!65}a^{5}-\frac{30\!\cdots\!83}{16\!\cdots\!65}a^{4}+\frac{70\!\cdots\!16}{16\!\cdots\!65}a^{3}+\frac{45\!\cdots\!71}{16\!\cdots\!65}a^{2}-\frac{26\!\cdots\!07}{323617895320073}a-\frac{745651396612569}{323617895320073}$, $\frac{12\!\cdots\!07}{16\!\cdots\!65}a^{20}-\frac{62\!\cdots\!69}{16\!\cdots\!65}a^{19}-\frac{29\!\cdots\!49}{16\!\cdots\!65}a^{18}+\frac{20\!\cdots\!58}{16\!\cdots\!65}a^{17}+\frac{85\!\cdots\!77}{16\!\cdots\!65}a^{16}-\frac{22\!\cdots\!57}{16\!\cdots\!65}a^{15}+\frac{21\!\cdots\!52}{16\!\cdots\!65}a^{14}+\frac{10\!\cdots\!84}{16\!\cdots\!65}a^{13}-\frac{18\!\cdots\!14}{16\!\cdots\!65}a^{12}-\frac{36\!\cdots\!23}{323617895320073}a^{11}+\frac{60\!\cdots\!24}{16\!\cdots\!65}a^{10}-\frac{63\!\cdots\!83}{16\!\cdots\!65}a^{9}-\frac{77\!\cdots\!98}{16\!\cdots\!65}a^{8}+\frac{47\!\cdots\!97}{16\!\cdots\!65}a^{7}+\frac{28\!\cdots\!59}{16\!\cdots\!65}a^{6}-\frac{33\!\cdots\!78}{16\!\cdots\!65}a^{5}+\frac{14\!\cdots\!76}{323617895320073}a^{4}+\frac{66\!\cdots\!52}{16\!\cdots\!65}a^{3}-\frac{26\!\cdots\!29}{16\!\cdots\!65}a^{2}+\frac{81\!\cdots\!66}{16\!\cdots\!65}a+\frac{65\!\cdots\!16}{16\!\cdots\!65}$, $\frac{36\!\cdots\!39}{323617895320073}a^{20}-\frac{17\!\cdots\!67}{323617895320073}a^{19}-\frac{42\!\cdots\!36}{16\!\cdots\!65}a^{18}+\frac{28\!\cdots\!07}{16\!\cdots\!65}a^{17}+\frac{12\!\cdots\!16}{16\!\cdots\!65}a^{16}-\frac{31\!\cdots\!97}{16\!\cdots\!65}a^{15}+\frac{57\!\cdots\!89}{323617895320073}a^{14}+\frac{14\!\cdots\!54}{16\!\cdots\!65}a^{13}-\frac{26\!\cdots\!31}{16\!\cdots\!65}a^{12}-\frac{26\!\cdots\!13}{16\!\cdots\!65}a^{11}+\frac{83\!\cdots\!96}{16\!\cdots\!65}a^{10}-\frac{71\!\cdots\!63}{16\!\cdots\!65}a^{9}-\frac{10\!\cdots\!01}{16\!\cdots\!65}a^{8}+\frac{63\!\cdots\!46}{16\!\cdots\!65}a^{7}+\frac{40\!\cdots\!88}{16\!\cdots\!65}a^{6}-\frac{89\!\cdots\!71}{323617895320073}a^{5}+\frac{92\!\cdots\!11}{16\!\cdots\!65}a^{4}+\frac{91\!\cdots\!49}{16\!\cdots\!65}a^{3}-\frac{34\!\cdots\!29}{16\!\cdots\!65}a^{2}+\frac{12\!\cdots\!68}{16\!\cdots\!65}a+\frac{87\!\cdots\!68}{16\!\cdots\!65}$, $\frac{45\!\cdots\!18}{16\!\cdots\!65}a^{20}-\frac{21\!\cdots\!63}{16\!\cdots\!65}a^{19}-\frac{10\!\cdots\!84}{16\!\cdots\!65}a^{18}+\frac{71\!\cdots\!68}{16\!\cdots\!65}a^{17}+\frac{32\!\cdots\!21}{16\!\cdots\!65}a^{16}-\frac{78\!\cdots\!67}{16\!\cdots\!65}a^{15}+\frac{71\!\cdots\!46}{16\!\cdots\!65}a^{14}+\frac{36\!\cdots\!72}{16\!\cdots\!65}a^{13}-\frac{65\!\cdots\!14}{16\!\cdots\!65}a^{12}-\frac{66\!\cdots\!54}{16\!\cdots\!65}a^{11}+\frac{20\!\cdots\!97}{16\!\cdots\!65}a^{10}-\frac{31\!\cdots\!05}{323617895320073}a^{9}-\frac{27\!\cdots\!72}{16\!\cdots\!65}a^{8}+\frac{15\!\cdots\!46}{16\!\cdots\!65}a^{7}+\frac{10\!\cdots\!62}{16\!\cdots\!65}a^{6}-\frac{22\!\cdots\!02}{323617895320073}a^{5}+\frac{43\!\cdots\!15}{323617895320073}a^{4}+\frac{25\!\cdots\!23}{16\!\cdots\!65}a^{3}-\frac{81\!\cdots\!83}{16\!\cdots\!65}a^{2}+\frac{23\!\cdots\!91}{16\!\cdots\!65}a+\frac{42\!\cdots\!79}{323617895320073}$, $\frac{46\!\cdots\!22}{16\!\cdots\!65}a^{20}-\frac{21\!\cdots\!31}{16\!\cdots\!65}a^{19}-\frac{11\!\cdots\!87}{16\!\cdots\!65}a^{18}+\frac{72\!\cdots\!29}{16\!\cdots\!65}a^{17}+\frac{93\!\cdots\!64}{323617895320073}a^{16}-\frac{80\!\cdots\!76}{16\!\cdots\!65}a^{15}+\frac{11\!\cdots\!56}{323617895320073}a^{14}+\frac{79\!\cdots\!64}{323617895320073}a^{13}-\frac{60\!\cdots\!07}{16\!\cdots\!65}a^{12}-\frac{81\!\cdots\!74}{16\!\cdots\!65}a^{11}+\frac{40\!\cdots\!99}{323617895320073}a^{10}+\frac{23\!\cdots\!94}{16\!\cdots\!65}a^{9}-\frac{28\!\cdots\!63}{16\!\cdots\!65}a^{8}+\frac{21\!\cdots\!02}{323617895320073}a^{7}+\frac{26\!\cdots\!39}{323617895320073}a^{6}-\frac{92\!\cdots\!96}{16\!\cdots\!65}a^{5}+\frac{87\!\cdots\!06}{323617895320073}a^{4}+\frac{40\!\cdots\!12}{16\!\cdots\!65}a^{3}-\frac{22\!\cdots\!81}{16\!\cdots\!65}a^{2}-\frac{24\!\cdots\!72}{16\!\cdots\!65}a+\frac{123865274934137}{16\!\cdots\!65}$, $\frac{72\!\cdots\!16}{323617895320073}a^{20}-\frac{17\!\cdots\!07}{16\!\cdots\!65}a^{19}-\frac{84\!\cdots\!22}{16\!\cdots\!65}a^{18}+\frac{57\!\cdots\!99}{16\!\cdots\!65}a^{17}+\frac{25\!\cdots\!62}{16\!\cdots\!65}a^{16}-\frac{62\!\cdots\!07}{16\!\cdots\!65}a^{15}+\frac{57\!\cdots\!23}{16\!\cdots\!65}a^{14}+\frac{29\!\cdots\!37}{16\!\cdots\!65}a^{13}-\frac{52\!\cdots\!12}{16\!\cdots\!65}a^{12}-\frac{53\!\cdots\!96}{16\!\cdots\!65}a^{11}+\frac{33\!\cdots\!78}{323617895320073}a^{10}-\frac{26\!\cdots\!57}{323617895320073}a^{9}-\frac{21\!\cdots\!04}{16\!\cdots\!65}a^{8}+\frac{12\!\cdots\!32}{16\!\cdots\!65}a^{7}+\frac{83\!\cdots\!03}{16\!\cdots\!65}a^{6}-\frac{90\!\cdots\!98}{16\!\cdots\!65}a^{5}+\frac{17\!\cdots\!24}{16\!\cdots\!65}a^{4}+\frac{22\!\cdots\!61}{16\!\cdots\!65}a^{3}-\frac{68\!\cdots\!53}{16\!\cdots\!65}a^{2}+\frac{12\!\cdots\!64}{16\!\cdots\!65}a+\frac{25\!\cdots\!33}{16\!\cdots\!65}$, $\frac{10\!\cdots\!64}{16\!\cdots\!65}a^{20}-\frac{51\!\cdots\!08}{16\!\cdots\!65}a^{19}-\frac{24\!\cdots\!87}{16\!\cdots\!65}a^{18}+\frac{16\!\cdots\!94}{16\!\cdots\!65}a^{17}+\frac{76\!\cdots\!48}{16\!\cdots\!65}a^{16}-\frac{18\!\cdots\!62}{16\!\cdots\!65}a^{15}+\frac{16\!\cdots\!99}{16\!\cdots\!65}a^{14}+\frac{86\!\cdots\!24}{16\!\cdots\!65}a^{13}-\frac{15\!\cdots\!72}{16\!\cdots\!65}a^{12}-\frac{15\!\cdots\!42}{16\!\cdots\!65}a^{11}+\frac{48\!\cdots\!17}{16\!\cdots\!65}a^{10}-\frac{43\!\cdots\!03}{16\!\cdots\!65}a^{9}-\frac{12\!\cdots\!80}{323617895320073}a^{8}+\frac{37\!\cdots\!78}{16\!\cdots\!65}a^{7}+\frac{46\!\cdots\!77}{323617895320073}a^{6}-\frac{26\!\cdots\!46}{16\!\cdots\!65}a^{5}+\frac{56\!\cdots\!14}{16\!\cdots\!65}a^{4}+\frac{10\!\cdots\!82}{323617895320073}a^{3}-\frac{21\!\cdots\!39}{16\!\cdots\!65}a^{2}+\frac{76\!\cdots\!89}{16\!\cdots\!65}a+\frac{78\!\cdots\!84}{16\!\cdots\!65}$, $\frac{14\!\cdots\!74}{16\!\cdots\!65}a^{20}-\frac{71\!\cdots\!94}{16\!\cdots\!65}a^{19}-\frac{33\!\cdots\!08}{16\!\cdots\!65}a^{18}+\frac{23\!\cdots\!66}{16\!\cdots\!65}a^{17}+\frac{10\!\cdots\!29}{16\!\cdots\!65}a^{16}-\frac{25\!\cdots\!78}{16\!\cdots\!65}a^{15}+\frac{23\!\cdots\!33}{16\!\cdots\!65}a^{14}+\frac{23\!\cdots\!02}{323617895320073}a^{13}-\frac{21\!\cdots\!73}{16\!\cdots\!65}a^{12}-\frac{42\!\cdots\!19}{323617895320073}a^{11}+\frac{68\!\cdots\!82}{16\!\cdots\!65}a^{10}-\frac{57\!\cdots\!98}{16\!\cdots\!65}a^{9}-\frac{88\!\cdots\!27}{16\!\cdots\!65}a^{8}+\frac{51\!\cdots\!54}{16\!\cdots\!65}a^{7}+\frac{33\!\cdots\!24}{16\!\cdots\!65}a^{6}-\frac{73\!\cdots\!86}{323617895320073}a^{5}+\frac{72\!\cdots\!96}{16\!\cdots\!65}a^{4}+\frac{87\!\cdots\!38}{16\!\cdots\!65}a^{3}-\frac{27\!\cdots\!33}{16\!\cdots\!65}a^{2}+\frac{57\!\cdots\!76}{16\!\cdots\!65}a+\frac{68\!\cdots\!28}{16\!\cdots\!65}$, $\frac{852148576815825}{323617895320073}a^{20}-\frac{20\!\cdots\!44}{16\!\cdots\!65}a^{19}-\frac{99\!\cdots\!89}{16\!\cdots\!65}a^{18}+\frac{67\!\cdots\!88}{16\!\cdots\!65}a^{17}+\frac{29\!\cdots\!79}{16\!\cdots\!65}a^{16}-\frac{73\!\cdots\!64}{16\!\cdots\!65}a^{15}+\frac{68\!\cdots\!01}{16\!\cdots\!65}a^{14}+\frac{34\!\cdots\!64}{16\!\cdots\!65}a^{13}-\frac{61\!\cdots\!39}{16\!\cdots\!65}a^{12}-\frac{60\!\cdots\!22}{16\!\cdots\!65}a^{11}+\frac{39\!\cdots\!23}{323617895320073}a^{10}-\frac{44\!\cdots\!15}{323617895320073}a^{9}-\frac{25\!\cdots\!83}{16\!\cdots\!65}a^{8}+\frac{15\!\cdots\!84}{16\!\cdots\!65}a^{7}+\frac{88\!\cdots\!96}{16\!\cdots\!65}a^{6}-\frac{10\!\cdots\!06}{16\!\cdots\!65}a^{5}+\frac{25\!\cdots\!33}{16\!\cdots\!65}a^{4}+\frac{12\!\cdots\!22}{16\!\cdots\!65}a^{3}-\frac{10\!\cdots\!81}{16\!\cdots\!65}a^{2}+\frac{58\!\cdots\!38}{16\!\cdots\!65}a+\frac{43\!\cdots\!86}{16\!\cdots\!65}$, $\frac{11\!\cdots\!07}{16\!\cdots\!65}a^{20}-\frac{54\!\cdots\!64}{16\!\cdots\!65}a^{19}-\frac{54\!\cdots\!88}{323617895320073}a^{18}+\frac{35\!\cdots\!52}{323617895320073}a^{17}+\frac{94\!\cdots\!23}{16\!\cdots\!65}a^{16}-\frac{19\!\cdots\!29}{16\!\cdots\!65}a^{15}+\frac{16\!\cdots\!57}{16\!\cdots\!65}a^{14}+\frac{94\!\cdots\!93}{16\!\cdots\!65}a^{13}-\frac{31\!\cdots\!66}{323617895320073}a^{12}-\frac{17\!\cdots\!48}{16\!\cdots\!65}a^{11}+\frac{10\!\cdots\!94}{323617895320073}a^{10}-\frac{64\!\cdots\!66}{16\!\cdots\!65}a^{9}-\frac{68\!\cdots\!54}{16\!\cdots\!65}a^{8}+\frac{35\!\cdots\!68}{16\!\cdots\!65}a^{7}+\frac{28\!\cdots\!77}{16\!\cdots\!65}a^{6}-\frac{26\!\cdots\!68}{16\!\cdots\!65}a^{5}+\frac{39\!\cdots\!41}{16\!\cdots\!65}a^{4}+\frac{83\!\cdots\!81}{16\!\cdots\!65}a^{3}-\frac{15\!\cdots\!08}{16\!\cdots\!65}a^{2}-\frac{35\!\cdots\!06}{16\!\cdots\!65}a+\frac{18\!\cdots\!14}{16\!\cdots\!65}$, $\frac{28\!\cdots\!89}{16\!\cdots\!65}a^{20}-\frac{13\!\cdots\!93}{16\!\cdots\!65}a^{19}-\frac{66\!\cdots\!92}{16\!\cdots\!65}a^{18}+\frac{45\!\cdots\!54}{16\!\cdots\!65}a^{17}+\frac{19\!\cdots\!88}{16\!\cdots\!65}a^{16}-\frac{49\!\cdots\!82}{16\!\cdots\!65}a^{15}+\frac{45\!\cdots\!19}{16\!\cdots\!65}a^{14}+\frac{23\!\cdots\!14}{16\!\cdots\!65}a^{13}-\frac{41\!\cdots\!27}{16\!\cdots\!65}a^{12}-\frac{41\!\cdots\!52}{16\!\cdots\!65}a^{11}+\frac{13\!\cdots\!32}{16\!\cdots\!65}a^{10}-\frac{12\!\cdots\!53}{16\!\cdots\!65}a^{9}-\frac{34\!\cdots\!99}{323617895320073}a^{8}+\frac{10\!\cdots\!28}{16\!\cdots\!65}a^{7}+\frac{12\!\cdots\!44}{323617895320073}a^{6}-\frac{71\!\cdots\!91}{16\!\cdots\!65}a^{5}+\frac{14\!\cdots\!24}{16\!\cdots\!65}a^{4}+\frac{30\!\cdots\!93}{323617895320073}a^{3}-\frac{54\!\cdots\!39}{16\!\cdots\!65}a^{2}+\frac{17\!\cdots\!64}{16\!\cdots\!65}a+\frac{12\!\cdots\!54}{16\!\cdots\!65}$, $\frac{32\!\cdots\!04}{16\!\cdots\!65}a^{20}-\frac{16\!\cdots\!02}{16\!\cdots\!65}a^{19}-\frac{14\!\cdots\!52}{323617895320073}a^{18}+\frac{10\!\cdots\!22}{323617895320073}a^{17}+\frac{94\!\cdots\!76}{16\!\cdots\!65}a^{16}-\frac{57\!\cdots\!99}{16\!\cdots\!65}a^{15}+\frac{13\!\cdots\!87}{323617895320073}a^{14}+\frac{25\!\cdots\!04}{16\!\cdots\!65}a^{13}-\frac{10\!\cdots\!15}{323617895320073}a^{12}-\frac{36\!\cdots\!56}{16\!\cdots\!65}a^{11}+\frac{16\!\cdots\!81}{16\!\cdots\!65}a^{10}-\frac{97\!\cdots\!58}{323617895320073}a^{9}-\frac{19\!\cdots\!12}{16\!\cdots\!65}a^{8}+\frac{16\!\cdots\!11}{16\!\cdots\!65}a^{7}+\frac{49\!\cdots\!48}{16\!\cdots\!65}a^{6}-\frac{10\!\cdots\!27}{16\!\cdots\!65}a^{5}+\frac{35\!\cdots\!96}{16\!\cdots\!65}a^{4}-\frac{16\!\cdots\!07}{16\!\cdots\!65}a^{3}-\frac{10\!\cdots\!51}{16\!\cdots\!65}a^{2}+\frac{14\!\cdots\!14}{16\!\cdots\!65}a-\frac{12\!\cdots\!13}{16\!\cdots\!65}$, $\frac{11\!\cdots\!56}{16\!\cdots\!65}a^{20}-\frac{57\!\cdots\!27}{16\!\cdots\!65}a^{19}-\frac{27\!\cdots\!21}{16\!\cdots\!65}a^{18}+\frac{18\!\cdots\!37}{16\!\cdots\!65}a^{17}+\frac{15\!\cdots\!21}{323617895320073}a^{16}-\frac{20\!\cdots\!09}{16\!\cdots\!65}a^{15}+\frac{19\!\cdots\!51}{16\!\cdots\!65}a^{14}+\frac{96\!\cdots\!58}{16\!\cdots\!65}a^{13}-\frac{17\!\cdots\!06}{16\!\cdots\!65}a^{12}-\frac{17\!\cdots\!02}{16\!\cdots\!65}a^{11}+\frac{55\!\cdots\!46}{16\!\cdots\!65}a^{10}-\frac{57\!\cdots\!16}{16\!\cdots\!65}a^{9}-\frac{71\!\cdots\!18}{16\!\cdots\!65}a^{8}+\frac{87\!\cdots\!03}{323617895320073}a^{7}+\frac{26\!\cdots\!34}{16\!\cdots\!65}a^{6}-\frac{30\!\cdots\!09}{16\!\cdots\!65}a^{5}+\frac{64\!\cdots\!69}{16\!\cdots\!65}a^{4}+\frac{69\!\cdots\!07}{16\!\cdots\!65}a^{3}-\frac{25\!\cdots\!88}{16\!\cdots\!65}a^{2}+\frac{95\!\cdots\!80}{323617895320073}a+\frac{20\!\cdots\!80}{323617895320073}$, $\frac{12\!\cdots\!22}{16\!\cdots\!65}a^{20}-\frac{12\!\cdots\!31}{323617895320073}a^{19}-\frac{30\!\cdots\!63}{16\!\cdots\!65}a^{18}+\frac{20\!\cdots\!36}{16\!\cdots\!65}a^{17}+\frac{97\!\cdots\!26}{16\!\cdots\!65}a^{16}-\frac{22\!\cdots\!49}{16\!\cdots\!65}a^{15}+\frac{19\!\cdots\!41}{16\!\cdots\!65}a^{14}+\frac{10\!\cdots\!47}{16\!\cdots\!65}a^{13}-\frac{18\!\cdots\!33}{16\!\cdots\!65}a^{12}-\frac{19\!\cdots\!77}{16\!\cdots\!65}a^{11}+\frac{59\!\cdots\!37}{16\!\cdots\!65}a^{10}-\frac{27\!\cdots\!87}{16\!\cdots\!65}a^{9}-\frac{78\!\cdots\!13}{16\!\cdots\!65}a^{8}+\frac{42\!\cdots\!61}{16\!\cdots\!65}a^{7}+\frac{31\!\cdots\!07}{16\!\cdots\!65}a^{6}-\frac{31\!\cdots\!87}{16\!\cdots\!65}a^{5}+\frac{10\!\cdots\!06}{323617895320073}a^{4}+\frac{82\!\cdots\!22}{16\!\cdots\!65}a^{3}-\frac{37\!\cdots\!74}{323617895320073}a^{2}+\frac{23\!\cdots\!07}{16\!\cdots\!65}a+\frac{12\!\cdots\!59}{16\!\cdots\!65}$, $\frac{425181578901794}{16\!\cdots\!65}a^{20}-\frac{15\!\cdots\!41}{16\!\cdots\!65}a^{19}-\frac{12\!\cdots\!03}{16\!\cdots\!65}a^{18}+\frac{55\!\cdots\!31}{16\!\cdots\!65}a^{17}+\frac{11\!\cdots\!04}{16\!\cdots\!65}a^{16}-\frac{70\!\cdots\!26}{16\!\cdots\!65}a^{15}-\frac{26\!\cdots\!44}{16\!\cdots\!65}a^{14}+\frac{43\!\cdots\!59}{16\!\cdots\!65}a^{13}-\frac{17\!\cdots\!03}{16\!\cdots\!65}a^{12}-\frac{28\!\cdots\!21}{323617895320073}a^{11}+\frac{23\!\cdots\!49}{323617895320073}a^{10}+\frac{24\!\cdots\!78}{16\!\cdots\!65}a^{9}-\frac{26\!\cdots\!24}{16\!\cdots\!65}a^{8}-\frac{19\!\cdots\!96}{16\!\cdots\!65}a^{7}+\frac{27\!\cdots\!41}{16\!\cdots\!65}a^{6}+\frac{36\!\cdots\!67}{16\!\cdots\!65}a^{5}-\frac{11\!\cdots\!52}{16\!\cdots\!65}a^{4}+\frac{19\!\cdots\!56}{16\!\cdots\!65}a^{3}+\frac{37\!\cdots\!97}{16\!\cdots\!65}a^{2}-\frac{50\!\cdots\!96}{16\!\cdots\!65}a-\frac{860695873974072}{323617895320073}$, $\frac{16\!\cdots\!42}{16\!\cdots\!65}a^{20}-\frac{81\!\cdots\!44}{16\!\cdots\!65}a^{19}-\frac{39\!\cdots\!16}{16\!\cdots\!65}a^{18}+\frac{26\!\cdots\!07}{16\!\cdots\!65}a^{17}+\frac{12\!\cdots\!49}{16\!\cdots\!65}a^{16}-\frac{29\!\cdots\!71}{16\!\cdots\!65}a^{15}+\frac{26\!\cdots\!62}{16\!\cdots\!65}a^{14}+\frac{13\!\cdots\!22}{16\!\cdots\!65}a^{13}-\frac{24\!\cdots\!76}{16\!\cdots\!65}a^{12}-\frac{24\!\cdots\!81}{16\!\cdots\!65}a^{11}+\frac{77\!\cdots\!46}{16\!\cdots\!65}a^{10}-\frac{67\!\cdots\!04}{16\!\cdots\!65}a^{9}-\frac{20\!\cdots\!62}{323617895320073}a^{8}+\frac{59\!\cdots\!19}{16\!\cdots\!65}a^{7}+\frac{75\!\cdots\!77}{323617895320073}a^{6}-\frac{42\!\cdots\!68}{16\!\cdots\!65}a^{5}+\frac{87\!\cdots\!67}{16\!\cdots\!65}a^{4}+\frac{18\!\cdots\!43}{323617895320073}a^{3}-\frac{33\!\cdots\!42}{16\!\cdots\!65}a^{2}+\frac{88\!\cdots\!97}{16\!\cdots\!65}a+\frac{86\!\cdots\!32}{16\!\cdots\!65}$
| 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: | \( 802783691.084 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 802783691.084 \cdot 1}{2\cdot\sqrt{31993368160435329741277209296896}}\cr\approx \mathstrut & 0.148822457749 \end{aligned}\] (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 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1)
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 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1, 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 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1);
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 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1);
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 21 |
| The 5 conjugacy class representatives for $C_7:C_3$ |
| Character table for $C_7:C_3$ |
Intermediate fields
| 3.3.5329.1, 7.7.1817487424.1 x7 |
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 7 sibling: | 7.7.1817487424.1 |
| Minimal sibling: | 7.7.1817487424.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }^{7}$ | ${\href{/padicField/7.7.0.1}{7} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.3.0.1}{3} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.3.0.1}{3} }^{7}$ | ${\href{/padicField/31.3.0.1}{3} }^{7}$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.3.0.1}{3} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{7}$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.18.1 | $x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 62 x^{15} + 111 x^{14} + 98 x^{13} - 189 x^{12} - 189 x^{11} + 259 x^{10} + 1496 x^{9} + 2586 x^{8} + 925 x^{7} + 798 x^{6} - 1092 x^{5} + 1029 x^{4} - 174 x^{3} - 53 x^{2} - 313 x + 131$ | $7$ | $3$ | $18$ | 21T2 | $[\ ]_{7}^{3}$ |
|
\(73\)
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |