Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*x - 1)
gp: K = bnfinit(y^21 - 7*y^20 - 42*y^19 + 476*y^18 - 630*y^17 - 6384*y^16 + 26810*y^15 - 16603*y^14 - 134918*y^13 + 424998*y^12 - 570745*y^11 + 331016*y^10 + 63070*y^9 - 212492*y^8 + 109569*y^7 - 1302*y^6 - 18788*y^5 + 6559*y^4 - 462*y^3 - 154*y^2 + 28*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*x - 1)
\( x^{21} - 7 x^{20} - 42 x^{19} + 476 x^{18} - 630 x^{17} - 6384 x^{16} + 26810 x^{15} - 16603 x^{14} + \cdots - 1 \)
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: |
\(656939336998075042895784450637466521\)
\(\medspace = 7^{32}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.77\) | 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: | $7^{236/147}29^{6/7}\approx 407.6054997453894$ | ||
| Ramified primes: |
\(7\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{3}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{12}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{5}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{7}-\frac{3}{7}$, $\frac{1}{7}a^{15}+\frac{3}{7}a^{8}-\frac{3}{7}a$, $\frac{1}{7}a^{16}+\frac{3}{7}a^{9}-\frac{3}{7}a^{2}$, $\frac{1}{7}a^{17}+\frac{3}{7}a^{10}-\frac{3}{7}a^{3}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{497}a^{19}+\frac{34}{497}a^{18}+\frac{18}{497}a^{17}+\frac{20}{497}a^{16}+\frac{34}{497}a^{15}-\frac{4}{497}a^{14}+\frac{34}{497}a^{13}-\frac{4}{497}a^{12}-\frac{27}{497}a^{11}+\frac{35}{71}a^{10}+\frac{150}{497}a^{9}+\frac{41}{497}a^{8}+\frac{17}{71}a^{7}-\frac{33}{497}a^{6}+\frac{235}{497}a^{5}+\frac{170}{497}a^{4}-\frac{128}{497}a^{3}-\frac{44}{497}a^{2}+\frac{216}{497}a-\frac{194}{497}$, $\frac{1}{11\!\cdots\!51}a^{20}-\frac{39780275299035}{11\!\cdots\!51}a^{19}-\frac{10\!\cdots\!85}{11\!\cdots\!51}a^{18}-\frac{759140281659462}{16\!\cdots\!93}a^{17}-\frac{22\!\cdots\!61}{11\!\cdots\!51}a^{16}+\frac{25\!\cdots\!91}{11\!\cdots\!51}a^{15}-\frac{44\!\cdots\!38}{11\!\cdots\!51}a^{14}+\frac{10\!\cdots\!28}{16\!\cdots\!93}a^{13}-\frac{54\!\cdots\!06}{11\!\cdots\!51}a^{12}+\frac{44\!\cdots\!68}{11\!\cdots\!51}a^{11}+\frac{517243234115833}{11\!\cdots\!51}a^{10}+\frac{26\!\cdots\!59}{11\!\cdots\!51}a^{9}+\frac{35\!\cdots\!85}{11\!\cdots\!51}a^{8}-\frac{50\!\cdots\!48}{11\!\cdots\!51}a^{7}-\frac{32\!\cdots\!93}{11\!\cdots\!51}a^{6}+\frac{34\!\cdots\!78}{16\!\cdots\!93}a^{5}-\frac{47\!\cdots\!15}{11\!\cdots\!51}a^{4}-\frac{80\!\cdots\!46}{11\!\cdots\!51}a^{3}-\frac{63\!\cdots\!69}{16\!\cdots\!93}a^{2}-\frac{34\!\cdots\!86}{16\!\cdots\!93}a+\frac{32\!\cdots\!63}{11\!\cdots\!51}$
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: |
$\frac{21\!\cdots\!61}{11\!\cdots\!51}a^{20}-\frac{14\!\cdots\!41}{11\!\cdots\!51}a^{19}-\frac{95\!\cdots\!25}{11\!\cdots\!51}a^{18}+\frac{10\!\cdots\!87}{11\!\cdots\!51}a^{17}-\frac{11\!\cdots\!14}{11\!\cdots\!51}a^{16}-\frac{14\!\cdots\!91}{11\!\cdots\!51}a^{15}+\frac{55\!\cdots\!18}{11\!\cdots\!51}a^{14}-\frac{21\!\cdots\!24}{11\!\cdots\!51}a^{13}-\frac{30\!\cdots\!92}{11\!\cdots\!51}a^{12}+\frac{85\!\cdots\!06}{11\!\cdots\!51}a^{11}-\frac{10\!\cdots\!73}{11\!\cdots\!51}a^{10}+\frac{44\!\cdots\!96}{11\!\cdots\!51}a^{9}+\frac{26\!\cdots\!67}{11\!\cdots\!51}a^{8}-\frac{39\!\cdots\!90}{11\!\cdots\!51}a^{7}+\frac{13\!\cdots\!30}{11\!\cdots\!51}a^{6}+\frac{34\!\cdots\!76}{11\!\cdots\!51}a^{5}-\frac{32\!\cdots\!90}{11\!\cdots\!51}a^{4}+\frac{55\!\cdots\!47}{11\!\cdots\!51}a^{3}+\frac{53\!\cdots\!70}{11\!\cdots\!51}a^{2}-\frac{19\!\cdots\!61}{11\!\cdots\!51}a+\frac{75\!\cdots\!35}{11\!\cdots\!51}$, $\frac{21\!\cdots\!61}{11\!\cdots\!51}a^{20}-\frac{14\!\cdots\!41}{11\!\cdots\!51}a^{19}-\frac{95\!\cdots\!25}{11\!\cdots\!51}a^{18}+\frac{10\!\cdots\!87}{11\!\cdots\!51}a^{17}-\frac{11\!\cdots\!14}{11\!\cdots\!51}a^{16}-\frac{14\!\cdots\!91}{11\!\cdots\!51}a^{15}+\frac{55\!\cdots\!18}{11\!\cdots\!51}a^{14}-\frac{21\!\cdots\!24}{11\!\cdots\!51}a^{13}-\frac{30\!\cdots\!92}{11\!\cdots\!51}a^{12}+\frac{85\!\cdots\!06}{11\!\cdots\!51}a^{11}-\frac{10\!\cdots\!73}{11\!\cdots\!51}a^{10}+\frac{44\!\cdots\!96}{11\!\cdots\!51}a^{9}+\frac{26\!\cdots\!67}{11\!\cdots\!51}a^{8}-\frac{39\!\cdots\!90}{11\!\cdots\!51}a^{7}+\frac{13\!\cdots\!30}{11\!\cdots\!51}a^{6}+\frac{34\!\cdots\!76}{11\!\cdots\!51}a^{5}-\frac{32\!\cdots\!90}{11\!\cdots\!51}a^{4}+\frac{55\!\cdots\!47}{11\!\cdots\!51}a^{3}+\frac{53\!\cdots\!70}{11\!\cdots\!51}a^{2}-\frac{19\!\cdots\!61}{11\!\cdots\!51}a+\frac{77\!\cdots\!86}{11\!\cdots\!51}$, $\frac{18\!\cdots\!91}{11\!\cdots\!51}a^{20}-\frac{12\!\cdots\!83}{11\!\cdots\!51}a^{19}-\frac{80\!\cdots\!43}{11\!\cdots\!51}a^{18}+\frac{85\!\cdots\!73}{11\!\cdots\!51}a^{17}-\frac{91\!\cdots\!95}{11\!\cdots\!51}a^{16}-\frac{11\!\cdots\!45}{11\!\cdots\!51}a^{15}+\frac{45\!\cdots\!08}{11\!\cdots\!51}a^{14}-\frac{17\!\cdots\!40}{11\!\cdots\!51}a^{13}-\frac{25\!\cdots\!87}{11\!\cdots\!51}a^{12}+\frac{70\!\cdots\!35}{11\!\cdots\!51}a^{11}-\frac{84\!\cdots\!79}{11\!\cdots\!51}a^{10}+\frac{35\!\cdots\!83}{11\!\cdots\!51}a^{9}+\frac{22\!\cdots\!67}{11\!\cdots\!51}a^{8}-\frac{32\!\cdots\!33}{11\!\cdots\!51}a^{7}+\frac{10\!\cdots\!78}{11\!\cdots\!51}a^{6}+\frac{29\!\cdots\!59}{11\!\cdots\!51}a^{5}-\frac{25\!\cdots\!43}{11\!\cdots\!51}a^{4}+\frac{41\!\cdots\!44}{11\!\cdots\!51}a^{3}+\frac{45\!\cdots\!64}{11\!\cdots\!51}a^{2}-\frac{15\!\cdots\!77}{11\!\cdots\!51}a+\frac{62\!\cdots\!92}{11\!\cdots\!51}$, $\frac{19\!\cdots\!56}{11\!\cdots\!51}a^{20}-\frac{13\!\cdots\!82}{11\!\cdots\!51}a^{19}-\frac{12\!\cdots\!42}{16\!\cdots\!93}a^{18}+\frac{90\!\cdots\!24}{11\!\cdots\!51}a^{17}-\frac{94\!\cdots\!19}{11\!\cdots\!51}a^{16}-\frac{12\!\cdots\!34}{11\!\cdots\!51}a^{15}+\frac{48\!\cdots\!87}{11\!\cdots\!51}a^{14}-\frac{16\!\cdots\!71}{11\!\cdots\!51}a^{13}-\frac{26\!\cdots\!56}{11\!\cdots\!51}a^{12}+\frac{10\!\cdots\!74}{16\!\cdots\!93}a^{11}-\frac{87\!\cdots\!92}{11\!\cdots\!51}a^{10}+\frac{36\!\cdots\!55}{11\!\cdots\!51}a^{9}+\frac{23\!\cdots\!63}{11\!\cdots\!51}a^{8}-\frac{33\!\cdots\!77}{11\!\cdots\!51}a^{7}+\frac{10\!\cdots\!02}{11\!\cdots\!51}a^{6}+\frac{30\!\cdots\!17}{11\!\cdots\!51}a^{5}-\frac{38\!\cdots\!64}{16\!\cdots\!93}a^{4}+\frac{44\!\cdots\!58}{11\!\cdots\!51}a^{3}+\frac{45\!\cdots\!57}{11\!\cdots\!51}a^{2}-\frac{15\!\cdots\!37}{11\!\cdots\!51}a+\frac{62\!\cdots\!21}{11\!\cdots\!51}$, $\frac{13\!\cdots\!42}{11\!\cdots\!51}a^{20}-\frac{87\!\cdots\!96}{11\!\cdots\!51}a^{19}-\frac{59\!\cdots\!30}{11\!\cdots\!51}a^{18}+\frac{60\!\cdots\!26}{11\!\cdots\!51}a^{17}-\frac{58\!\cdots\!58}{11\!\cdots\!51}a^{16}-\frac{87\!\cdots\!92}{11\!\cdots\!51}a^{15}+\frac{32\!\cdots\!56}{11\!\cdots\!51}a^{14}-\frac{91\!\cdots\!10}{11\!\cdots\!51}a^{13}-\frac{18\!\cdots\!68}{11\!\cdots\!51}a^{12}+\frac{49\!\cdots\!18}{11\!\cdots\!51}a^{11}-\frac{56\!\cdots\!48}{11\!\cdots\!51}a^{10}+\frac{21\!\cdots\!54}{11\!\cdots\!51}a^{9}+\frac{16\!\cdots\!87}{11\!\cdots\!51}a^{8}-\frac{21\!\cdots\!03}{11\!\cdots\!51}a^{7}+\frac{61\!\cdots\!13}{11\!\cdots\!51}a^{6}+\frac{21\!\cdots\!31}{11\!\cdots\!51}a^{5}-\frac{16\!\cdots\!89}{11\!\cdots\!51}a^{4}+\frac{24\!\cdots\!45}{11\!\cdots\!51}a^{3}+\frac{30\!\cdots\!41}{11\!\cdots\!51}a^{2}-\frac{88\!\cdots\!83}{11\!\cdots\!51}a+\frac{35\!\cdots\!51}{11\!\cdots\!51}$, $\frac{26\!\cdots\!78}{11\!\cdots\!51}a^{20}-\frac{17\!\cdots\!06}{11\!\cdots\!51}a^{19}-\frac{11\!\cdots\!22}{11\!\cdots\!51}a^{18}+\frac{12\!\cdots\!96}{11\!\cdots\!51}a^{17}-\frac{12\!\cdots\!02}{11\!\cdots\!51}a^{16}-\frac{17\!\cdots\!39}{11\!\cdots\!51}a^{15}+\frac{92\!\cdots\!05}{16\!\cdots\!93}a^{14}-\frac{24\!\cdots\!63}{11\!\cdots\!51}a^{13}-\frac{35\!\cdots\!72}{11\!\cdots\!51}a^{12}+\frac{10\!\cdots\!11}{11\!\cdots\!51}a^{11}-\frac{11\!\cdots\!70}{11\!\cdots\!51}a^{10}+\frac{51\!\cdots\!38}{11\!\cdots\!51}a^{9}+\frac{31\!\cdots\!31}{11\!\cdots\!51}a^{8}-\frac{65\!\cdots\!78}{16\!\cdots\!93}a^{7}+\frac{15\!\cdots\!74}{11\!\cdots\!51}a^{6}+\frac{40\!\cdots\!03}{11\!\cdots\!51}a^{5}-\frac{36\!\cdots\!99}{11\!\cdots\!51}a^{4}+\frac{63\!\cdots\!75}{11\!\cdots\!51}a^{3}+\frac{61\!\cdots\!72}{11\!\cdots\!51}a^{2}-\frac{22\!\cdots\!05}{11\!\cdots\!51}a+\frac{12\!\cdots\!24}{16\!\cdots\!93}$, $\frac{29\!\cdots\!98}{11\!\cdots\!51}a^{20}-\frac{19\!\cdots\!85}{11\!\cdots\!51}a^{19}-\frac{12\!\cdots\!91}{11\!\cdots\!51}a^{18}+\frac{13\!\cdots\!27}{11\!\cdots\!51}a^{17}-\frac{14\!\cdots\!35}{11\!\cdots\!51}a^{16}-\frac{19\!\cdots\!11}{11\!\cdots\!51}a^{15}+\frac{73\!\cdots\!24}{11\!\cdots\!51}a^{14}-\frac{29\!\cdots\!76}{11\!\cdots\!51}a^{13}-\frac{40\!\cdots\!78}{11\!\cdots\!51}a^{12}+\frac{11\!\cdots\!08}{11\!\cdots\!51}a^{11}-\frac{13\!\cdots\!51}{11\!\cdots\!51}a^{10}+\frac{59\!\cdots\!95}{11\!\cdots\!51}a^{9}+\frac{35\!\cdots\!74}{11\!\cdots\!51}a^{8}-\frac{53\!\cdots\!70}{11\!\cdots\!51}a^{7}+\frac{17\!\cdots\!34}{11\!\cdots\!51}a^{6}+\frac{45\!\cdots\!14}{11\!\cdots\!51}a^{5}-\frac{42\!\cdots\!55}{11\!\cdots\!51}a^{4}+\frac{73\!\cdots\!69}{11\!\cdots\!51}a^{3}+\frac{70\!\cdots\!16}{11\!\cdots\!51}a^{2}-\frac{25\!\cdots\!56}{11\!\cdots\!51}a+\frac{10\!\cdots\!69}{11\!\cdots\!51}$, $\frac{21\!\cdots\!58}{11\!\cdots\!51}a^{20}-\frac{13\!\cdots\!62}{11\!\cdots\!51}a^{19}-\frac{93\!\cdots\!66}{11\!\cdots\!51}a^{18}+\frac{96\!\cdots\!44}{11\!\cdots\!51}a^{17}-\frac{96\!\cdots\!16}{11\!\cdots\!51}a^{16}-\frac{13\!\cdots\!66}{11\!\cdots\!51}a^{15}+\frac{51\!\cdots\!52}{11\!\cdots\!51}a^{14}-\frac{15\!\cdots\!34}{11\!\cdots\!51}a^{13}-\frac{29\!\cdots\!42}{11\!\cdots\!51}a^{12}+\frac{78\!\cdots\!14}{11\!\cdots\!51}a^{11}-\frac{91\!\cdots\!34}{11\!\cdots\!51}a^{10}+\frac{35\!\cdots\!98}{11\!\cdots\!51}a^{9}+\frac{26\!\cdots\!67}{11\!\cdots\!51}a^{8}-\frac{34\!\cdots\!10}{11\!\cdots\!51}a^{7}+\frac{10\!\cdots\!84}{11\!\cdots\!51}a^{6}+\frac{35\!\cdots\!82}{11\!\cdots\!51}a^{5}-\frac{26\!\cdots\!80}{11\!\cdots\!51}a^{4}+\frac{37\!\cdots\!66}{11\!\cdots\!51}a^{3}+\frac{51\!\cdots\!04}{11\!\cdots\!51}a^{2}-\frac{14\!\cdots\!76}{11\!\cdots\!51}a+\frac{49\!\cdots\!17}{11\!\cdots\!51}$, $\frac{80\!\cdots\!68}{11\!\cdots\!51}a^{20}-\frac{72\!\cdots\!16}{16\!\cdots\!93}a^{19}-\frac{52\!\cdots\!93}{16\!\cdots\!93}a^{18}+\frac{35\!\cdots\!04}{11\!\cdots\!51}a^{17}-\frac{27\!\cdots\!25}{11\!\cdots\!51}a^{16}-\frac{75\!\cdots\!71}{16\!\cdots\!81}a^{15}+\frac{18\!\cdots\!61}{11\!\cdots\!51}a^{14}-\frac{19\!\cdots\!13}{16\!\cdots\!81}a^{13}-\frac{11\!\cdots\!95}{11\!\cdots\!51}a^{12}+\frac{38\!\cdots\!77}{16\!\cdots\!93}a^{11}-\frac{39\!\cdots\!96}{16\!\cdots\!93}a^{10}+\frac{66\!\cdots\!82}{11\!\cdots\!51}a^{9}+\frac{11\!\cdots\!97}{11\!\cdots\!51}a^{8}-\frac{10\!\cdots\!48}{11\!\cdots\!51}a^{7}+\frac{12\!\cdots\!71}{11\!\cdots\!51}a^{6}+\frac{16\!\cdots\!74}{11\!\cdots\!51}a^{5}-\frac{87\!\cdots\!78}{16\!\cdots\!93}a^{4}-\frac{48\!\cdots\!04}{11\!\cdots\!51}a^{3}+\frac{19\!\cdots\!61}{11\!\cdots\!51}a^{2}-\frac{66\!\cdots\!19}{11\!\cdots\!51}a-\frac{14\!\cdots\!92}{11\!\cdots\!51}$, $\frac{34\!\cdots\!94}{11\!\cdots\!51}a^{20}-\frac{23\!\cdots\!95}{11\!\cdots\!51}a^{19}-\frac{15\!\cdots\!95}{11\!\cdots\!51}a^{18}+\frac{16\!\cdots\!74}{11\!\cdots\!51}a^{17}-\frac{23\!\cdots\!60}{16\!\cdots\!93}a^{16}-\frac{32\!\cdots\!85}{16\!\cdots\!93}a^{15}+\frac{85\!\cdots\!62}{11\!\cdots\!51}a^{14}-\frac{29\!\cdots\!77}{11\!\cdots\!51}a^{13}-\frac{47\!\cdots\!56}{11\!\cdots\!51}a^{12}+\frac{13\!\cdots\!58}{11\!\cdots\!51}a^{11}-\frac{22\!\cdots\!04}{16\!\cdots\!93}a^{10}+\frac{64\!\cdots\!79}{11\!\cdots\!51}a^{9}+\frac{42\!\cdots\!81}{11\!\cdots\!51}a^{8}-\frac{85\!\cdots\!00}{16\!\cdots\!93}a^{7}+\frac{18\!\cdots\!44}{11\!\cdots\!51}a^{6}+\frac{56\!\cdots\!01}{11\!\cdots\!51}a^{5}-\frac{66\!\cdots\!35}{16\!\cdots\!93}a^{4}+\frac{73\!\cdots\!62}{11\!\cdots\!51}a^{3}+\frac{81\!\cdots\!87}{11\!\cdots\!51}a^{2}-\frac{25\!\cdots\!52}{11\!\cdots\!51}a+\frac{10\!\cdots\!87}{11\!\cdots\!51}$, $\frac{59\!\cdots\!67}{11\!\cdots\!51}a^{20}-\frac{39\!\cdots\!85}{11\!\cdots\!51}a^{19}-\frac{25\!\cdots\!49}{11\!\cdots\!51}a^{18}+\frac{27\!\cdots\!93}{11\!\cdots\!51}a^{17}-\frac{29\!\cdots\!25}{11\!\cdots\!51}a^{16}-\frac{38\!\cdots\!79}{11\!\cdots\!51}a^{15}+\frac{21\!\cdots\!53}{16\!\cdots\!93}a^{14}-\frac{55\!\cdots\!26}{11\!\cdots\!51}a^{13}-\frac{81\!\cdots\!98}{11\!\cdots\!51}a^{12}+\frac{32\!\cdots\!32}{16\!\cdots\!93}a^{11}-\frac{27\!\cdots\!28}{11\!\cdots\!51}a^{10}+\frac{11\!\cdots\!76}{11\!\cdots\!51}a^{9}+\frac{71\!\cdots\!47}{11\!\cdots\!51}a^{8}-\frac{10\!\cdots\!81}{11\!\cdots\!51}a^{7}+\frac{48\!\cdots\!15}{16\!\cdots\!93}a^{6}+\frac{92\!\cdots\!13}{11\!\cdots\!51}a^{5}-\frac{84\!\cdots\!18}{11\!\cdots\!51}a^{4}+\frac{14\!\cdots\!15}{11\!\cdots\!51}a^{3}+\frac{14\!\cdots\!09}{11\!\cdots\!51}a^{2}-\frac{49\!\cdots\!74}{11\!\cdots\!51}a+\frac{19\!\cdots\!10}{11\!\cdots\!51}$, $\frac{12\!\cdots\!50}{11\!\cdots\!51}a^{20}-\frac{82\!\cdots\!63}{11\!\cdots\!51}a^{19}-\frac{55\!\cdots\!18}{11\!\cdots\!51}a^{18}+\frac{81\!\cdots\!10}{16\!\cdots\!93}a^{17}-\frac{57\!\cdots\!87}{11\!\cdots\!51}a^{16}-\frac{81\!\cdots\!38}{11\!\cdots\!51}a^{15}+\frac{30\!\cdots\!93}{11\!\cdots\!51}a^{14}-\frac{97\!\cdots\!32}{11\!\cdots\!51}a^{13}-\frac{17\!\cdots\!53}{11\!\cdots\!51}a^{12}+\frac{46\!\cdots\!50}{11\!\cdots\!51}a^{11}-\frac{54\!\cdots\!63}{11\!\cdots\!51}a^{10}+\frac{22\!\cdots\!89}{11\!\cdots\!51}a^{9}+\frac{15\!\cdots\!34}{11\!\cdots\!51}a^{8}-\frac{20\!\cdots\!15}{11\!\cdots\!51}a^{7}+\frac{64\!\cdots\!76}{11\!\cdots\!51}a^{6}+\frac{19\!\cdots\!84}{11\!\cdots\!51}a^{5}-\frac{16\!\cdots\!57}{11\!\cdots\!51}a^{4}+\frac{26\!\cdots\!39}{11\!\cdots\!51}a^{3}+\frac{27\!\cdots\!04}{11\!\cdots\!51}a^{2}-\frac{94\!\cdots\!61}{11\!\cdots\!51}a+\frac{37\!\cdots\!65}{11\!\cdots\!51}$, $\frac{10\!\cdots\!94}{16\!\cdots\!93}a^{20}-\frac{48\!\cdots\!42}{11\!\cdots\!51}a^{19}-\frac{33\!\cdots\!67}{11\!\cdots\!51}a^{18}+\frac{48\!\cdots\!49}{16\!\cdots\!93}a^{17}-\frac{32\!\cdots\!49}{11\!\cdots\!51}a^{16}-\frac{48\!\cdots\!90}{11\!\cdots\!51}a^{15}+\frac{17\!\cdots\!71}{11\!\cdots\!51}a^{14}-\frac{49\!\cdots\!18}{11\!\cdots\!51}a^{13}-\frac{10\!\cdots\!20}{11\!\cdots\!51}a^{12}+\frac{27\!\cdots\!15}{11\!\cdots\!51}a^{11}-\frac{31\!\cdots\!80}{11\!\cdots\!51}a^{10}+\frac{11\!\cdots\!89}{11\!\cdots\!51}a^{9}+\frac{94\!\cdots\!84}{11\!\cdots\!51}a^{8}-\frac{16\!\cdots\!20}{16\!\cdots\!93}a^{7}+\frac{32\!\cdots\!19}{11\!\cdots\!51}a^{6}+\frac{12\!\cdots\!00}{11\!\cdots\!51}a^{5}-\frac{12\!\cdots\!85}{16\!\cdots\!93}a^{4}+\frac{11\!\cdots\!59}{11\!\cdots\!51}a^{3}+\frac{16\!\cdots\!80}{11\!\cdots\!51}a^{2}-\frac{41\!\cdots\!97}{11\!\cdots\!51}a+\frac{14\!\cdots\!69}{11\!\cdots\!51}$, $\frac{45\!\cdots\!28}{11\!\cdots\!51}a^{20}-\frac{30\!\cdots\!85}{11\!\cdots\!51}a^{19}-\frac{19\!\cdots\!19}{11\!\cdots\!51}a^{18}+\frac{21\!\cdots\!29}{11\!\cdots\!51}a^{17}-\frac{22\!\cdots\!85}{11\!\cdots\!51}a^{16}-\frac{29\!\cdots\!87}{11\!\cdots\!51}a^{15}+\frac{11\!\cdots\!11}{11\!\cdots\!51}a^{14}-\frac{44\!\cdots\!63}{11\!\cdots\!51}a^{13}-\frac{88\!\cdots\!19}{16\!\cdots\!81}a^{12}+\frac{17\!\cdots\!31}{11\!\cdots\!51}a^{11}-\frac{21\!\cdots\!89}{11\!\cdots\!51}a^{10}+\frac{91\!\cdots\!99}{11\!\cdots\!51}a^{9}+\frac{54\!\cdots\!26}{11\!\cdots\!51}a^{8}-\frac{81\!\cdots\!92}{11\!\cdots\!51}a^{7}+\frac{26\!\cdots\!24}{11\!\cdots\!51}a^{6}+\frac{70\!\cdots\!93}{11\!\cdots\!51}a^{5}-\frac{65\!\cdots\!26}{11\!\cdots\!51}a^{4}+\frac{11\!\cdots\!24}{11\!\cdots\!51}a^{3}+\frac{10\!\cdots\!39}{11\!\cdots\!51}a^{2}-\frac{39\!\cdots\!97}{11\!\cdots\!51}a+\frac{23\!\cdots\!04}{16\!\cdots\!93}$, $\frac{10\!\cdots\!74}{16\!\cdots\!93}a^{20}-\frac{49\!\cdots\!86}{11\!\cdots\!51}a^{19}-\frac{32\!\cdots\!25}{11\!\cdots\!51}a^{18}+\frac{48\!\cdots\!72}{16\!\cdots\!93}a^{17}-\frac{35\!\cdots\!81}{11\!\cdots\!51}a^{16}-\frac{48\!\cdots\!83}{11\!\cdots\!51}a^{15}+\frac{18\!\cdots\!88}{11\!\cdots\!51}a^{14}-\frac{91\!\cdots\!88}{16\!\cdots\!93}a^{13}-\frac{14\!\cdots\!96}{16\!\cdots\!93}a^{12}+\frac{28\!\cdots\!96}{11\!\cdots\!51}a^{11}-\frac{32\!\cdots\!06}{11\!\cdots\!51}a^{10}+\frac{19\!\cdots\!45}{16\!\cdots\!93}a^{9}+\frac{91\!\cdots\!82}{11\!\cdots\!51}a^{8}-\frac{12\!\cdots\!02}{11\!\cdots\!51}a^{7}+\frac{56\!\cdots\!00}{16\!\cdots\!93}a^{6}+\frac{11\!\cdots\!31}{11\!\cdots\!51}a^{5}-\frac{99\!\cdots\!76}{11\!\cdots\!51}a^{4}+\frac{15\!\cdots\!22}{11\!\cdots\!51}a^{3}+\frac{24\!\cdots\!83}{16\!\cdots\!81}a^{2}-\frac{56\!\cdots\!04}{11\!\cdots\!51}a+\frac{22\!\cdots\!09}{11\!\cdots\!51}$, $\frac{17\!\cdots\!19}{11\!\cdots\!51}a^{20}-\frac{11\!\cdots\!74}{11\!\cdots\!51}a^{19}-\frac{75\!\cdots\!79}{11\!\cdots\!51}a^{18}+\frac{81\!\cdots\!23}{11\!\cdots\!51}a^{17}-\frac{92\!\cdots\!13}{11\!\cdots\!51}a^{16}-\frac{11\!\cdots\!46}{11\!\cdots\!51}a^{15}+\frac{44\!\cdots\!56}{11\!\cdots\!51}a^{14}-\frac{18\!\cdots\!02}{11\!\cdots\!51}a^{13}-\frac{34\!\cdots\!08}{16\!\cdots\!93}a^{12}+\frac{68\!\cdots\!34}{11\!\cdots\!51}a^{11}-\frac{82\!\cdots\!51}{11\!\cdots\!51}a^{10}+\frac{36\!\cdots\!38}{11\!\cdots\!51}a^{9}+\frac{31\!\cdots\!23}{16\!\cdots\!93}a^{8}-\frac{32\!\cdots\!41}{11\!\cdots\!51}a^{7}+\frac{10\!\cdots\!46}{11\!\cdots\!51}a^{6}+\frac{29\!\cdots\!71}{11\!\cdots\!51}a^{5}-\frac{25\!\cdots\!96}{11\!\cdots\!51}a^{4}+\frac{59\!\cdots\!45}{16\!\cdots\!93}a^{3}+\frac{45\!\cdots\!54}{11\!\cdots\!51}a^{2}-\frac{14\!\cdots\!59}{11\!\cdots\!51}a+\frac{58\!\cdots\!87}{11\!\cdots\!51}$, $\frac{34\!\cdots\!70}{11\!\cdots\!51}a^{20}-\frac{23\!\cdots\!22}{11\!\cdots\!51}a^{19}-\frac{15\!\cdots\!40}{11\!\cdots\!51}a^{18}+\frac{16\!\cdots\!78}{11\!\cdots\!51}a^{17}-\frac{16\!\cdots\!21}{11\!\cdots\!51}a^{16}-\frac{22\!\cdots\!79}{11\!\cdots\!51}a^{15}+\frac{86\!\cdots\!57}{11\!\cdots\!51}a^{14}-\frac{31\!\cdots\!57}{11\!\cdots\!51}a^{13}-\frac{47\!\cdots\!18}{11\!\cdots\!51}a^{12}+\frac{13\!\cdots\!90}{11\!\cdots\!51}a^{11}-\frac{15\!\cdots\!37}{11\!\cdots\!51}a^{10}+\frac{65\!\cdots\!91}{11\!\cdots\!51}a^{9}+\frac{43\!\cdots\!87}{11\!\cdots\!51}a^{8}-\frac{86\!\cdots\!57}{16\!\cdots\!93}a^{7}+\frac{18\!\cdots\!12}{11\!\cdots\!51}a^{6}+\frac{56\!\cdots\!51}{11\!\cdots\!51}a^{5}-\frac{47\!\cdots\!61}{11\!\cdots\!51}a^{4}+\frac{75\!\cdots\!68}{11\!\cdots\!51}a^{3}+\frac{86\!\cdots\!77}{11\!\cdots\!51}a^{2}-\frac{27\!\cdots\!36}{11\!\cdots\!51}a+\frac{10\!\cdots\!22}{11\!\cdots\!51}$, $\frac{34\!\cdots\!58}{11\!\cdots\!51}a^{20}-\frac{23\!\cdots\!00}{11\!\cdots\!51}a^{19}-\frac{15\!\cdots\!83}{11\!\cdots\!51}a^{18}+\frac{16\!\cdots\!80}{11\!\cdots\!51}a^{17}-\frac{26\!\cdots\!04}{16\!\cdots\!93}a^{16}-\frac{22\!\cdots\!05}{11\!\cdots\!51}a^{15}+\frac{12\!\cdots\!05}{16\!\cdots\!93}a^{14}-\frac{38\!\cdots\!81}{11\!\cdots\!51}a^{13}-\frac{69\!\cdots\!68}{16\!\cdots\!93}a^{12}+\frac{13\!\cdots\!29}{11\!\cdots\!51}a^{11}-\frac{16\!\cdots\!12}{11\!\cdots\!51}a^{10}+\frac{74\!\cdots\!74}{11\!\cdots\!51}a^{9}+\frac{42\!\cdots\!61}{11\!\cdots\!51}a^{8}-\frac{65\!\cdots\!73}{11\!\cdots\!51}a^{7}+\frac{21\!\cdots\!67}{11\!\cdots\!51}a^{6}+\frac{54\!\cdots\!06}{11\!\cdots\!51}a^{5}-\frac{52\!\cdots\!51}{11\!\cdots\!51}a^{4}+\frac{92\!\cdots\!48}{11\!\cdots\!51}a^{3}+\frac{83\!\cdots\!58}{11\!\cdots\!51}a^{2}-\frac{31\!\cdots\!76}{11\!\cdots\!51}a+\frac{14\!\cdots\!38}{11\!\cdots\!51}$, $\frac{11\!\cdots\!32}{11\!\cdots\!51}a^{20}-\frac{73\!\cdots\!74}{11\!\cdots\!51}a^{19}-\frac{49\!\cdots\!56}{11\!\cdots\!51}a^{18}+\frac{50\!\cdots\!85}{11\!\cdots\!51}a^{17}-\frac{51\!\cdots\!93}{11\!\cdots\!51}a^{16}-\frac{72\!\cdots\!02}{11\!\cdots\!51}a^{15}+\frac{27\!\cdots\!80}{11\!\cdots\!51}a^{14}-\frac{89\!\cdots\!53}{11\!\cdots\!51}a^{13}-\frac{15\!\cdots\!43}{11\!\cdots\!51}a^{12}+\frac{41\!\cdots\!09}{11\!\cdots\!51}a^{11}-\frac{69\!\cdots\!11}{16\!\cdots\!93}a^{10}+\frac{20\!\cdots\!06}{11\!\cdots\!51}a^{9}+\frac{13\!\cdots\!33}{11\!\cdots\!51}a^{8}-\frac{18\!\cdots\!71}{11\!\cdots\!51}a^{7}+\frac{57\!\cdots\!15}{11\!\cdots\!51}a^{6}+\frac{18\!\cdots\!15}{11\!\cdots\!51}a^{5}-\frac{14\!\cdots\!34}{11\!\cdots\!51}a^{4}+\frac{22\!\cdots\!64}{11\!\cdots\!51}a^{3}+\frac{28\!\cdots\!95}{11\!\cdots\!51}a^{2}-\frac{82\!\cdots\!13}{11\!\cdots\!51}a+\frac{27\!\cdots\!64}{11\!\cdots\!51}$, $\frac{11\!\cdots\!06}{11\!\cdots\!51}a^{20}-\frac{10\!\cdots\!57}{16\!\cdots\!93}a^{19}-\frac{49\!\cdots\!23}{11\!\cdots\!51}a^{18}+\frac{52\!\cdots\!58}{11\!\cdots\!51}a^{17}-\frac{58\!\cdots\!55}{11\!\cdots\!51}a^{16}-\frac{74\!\cdots\!38}{11\!\cdots\!51}a^{15}+\frac{28\!\cdots\!45}{11\!\cdots\!51}a^{14}-\frac{11\!\cdots\!76}{11\!\cdots\!51}a^{13}-\frac{15\!\cdots\!52}{11\!\cdots\!51}a^{12}+\frac{44\!\cdots\!06}{11\!\cdots\!51}a^{11}-\frac{53\!\cdots\!12}{11\!\cdots\!51}a^{10}+\frac{22\!\cdots\!08}{11\!\cdots\!51}a^{9}+\frac{14\!\cdots\!73}{11\!\cdots\!51}a^{8}-\frac{20\!\cdots\!36}{11\!\cdots\!51}a^{7}+\frac{66\!\cdots\!02}{11\!\cdots\!51}a^{6}+\frac{18\!\cdots\!23}{11\!\cdots\!51}a^{5}-\frac{16\!\cdots\!90}{11\!\cdots\!51}a^{4}+\frac{27\!\cdots\!36}{11\!\cdots\!51}a^{3}+\frac{28\!\cdots\!53}{11\!\cdots\!51}a^{2}-\frac{13\!\cdots\!84}{16\!\cdots\!93}a+\frac{36\!\cdots\!01}{11\!\cdots\!51}$
| 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: | \( 170660357100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 170660357100 \cdot 1}{2\cdot\sqrt{656939336998075042895784450637466521}}\cr\approx \mathstrut & 0.220785185578958 \end{aligned}\] (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 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*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 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*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 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*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 - 42*x^19 + 476*x^18 - 630*x^17 - 6384*x^16 + 26810*x^15 - 16603*x^14 - 134918*x^13 + 424998*x^12 - 570745*x^11 + 331016*x^10 + 63070*x^9 - 212492*x^8 + 109569*x^7 - 1302*x^6 - 18788*x^5 + 6559*x^4 - 462*x^3 - 154*x^2 + 28*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
$C_7\wr C_3$ (as 21T28):
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 1029 |
| The 133 conjugacy class representatives for $C_7\wr C_3$ are not computed |
| Character table for $C_7\wr C_3$ is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
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 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 | $21$ | $21$ | $21$ | R | $21$ | ${\href{/padicField/13.7.0.1}{7} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | $21$ | $21$ | R | $21$ | $21$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{7}$ | $21$ | $21$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $21$ | $21$ | $1$ | $32$ | |||
|
\(29\)
| 29.7.0.1 | $x^{7} + 2 x + 27$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 29.7.6.5 | $x^{7} + 58$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 29.7.0.1 | $x^{7} + 2 x + 27$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |