Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*x - 1)
gp: K = bnfinit(y^22 - 3*y^21 - 33*y^20 + 88*y^19 + 421*y^18 - 1001*y^17 - 2683*y^16 + 5647*y^15 + 9370*y^14 - 17186*y^13 - 18923*y^12 + 29509*y^11 + 23007*y^10 - 28981*y^9 - 17000*y^8 + 15763*y^7 + 7336*y^6 - 4249*y^5 - 1613*y^4 + 425*y^3 + 118*y^2 - 13*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*x - 1)
\( x^{22} - 3 x^{21} - 33 x^{20} + 88 x^{19} + 421 x^{18} - 1001 x^{17} - 2683 x^{16} + 5647 x^{15} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[22, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(115836063662590130514138078564453125\)
\(\medspace = 5^{11}\cdot 421^{2}\cdot 115692385433^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.25\) | 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: | $5^{1/2}421^{1/2}115692385433^{1/2}\approx 15605526.948375214$ | ||
| Ramified primes: |
\(5\), \(421\), \(115692385433\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{22\!\cdots\!07}a^{21}+\frac{84\!\cdots\!96}{22\!\cdots\!07}a^{20}+\frac{96\!\cdots\!10}{22\!\cdots\!07}a^{19}+\frac{38\!\cdots\!55}{22\!\cdots\!07}a^{18}+\frac{58\!\cdots\!33}{22\!\cdots\!07}a^{17}+\frac{44\!\cdots\!42}{22\!\cdots\!07}a^{16}-\frac{81\!\cdots\!77}{22\!\cdots\!07}a^{15}-\frac{97\!\cdots\!14}{22\!\cdots\!07}a^{14}+\frac{54\!\cdots\!76}{22\!\cdots\!07}a^{13}-\frac{87\!\cdots\!69}{22\!\cdots\!07}a^{12}-\frac{46\!\cdots\!26}{22\!\cdots\!07}a^{11}-\frac{20\!\cdots\!63}{22\!\cdots\!07}a^{10}+\frac{38\!\cdots\!07}{22\!\cdots\!07}a^{9}-\frac{95\!\cdots\!83}{22\!\cdots\!07}a^{8}+\frac{64\!\cdots\!13}{22\!\cdots\!07}a^{7}-\frac{17\!\cdots\!95}{22\!\cdots\!07}a^{6}+\frac{10\!\cdots\!69}{22\!\cdots\!07}a^{5}-\frac{17\!\cdots\!30}{22\!\cdots\!07}a^{4}+\frac{88\!\cdots\!38}{22\!\cdots\!07}a^{3}+\frac{11\!\cdots\!76}{22\!\cdots\!07}a^{2}-\frac{85\!\cdots\!08}{22\!\cdots\!07}a-\frac{60\!\cdots\!95}{22\!\cdots\!07}$
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: | $21$ | 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: |
$\frac{13\!\cdots\!61}{22\!\cdots\!07}a^{21}-\frac{14\!\cdots\!27}{22\!\cdots\!07}a^{20}-\frac{49\!\cdots\!68}{22\!\cdots\!07}a^{19}+\frac{29\!\cdots\!65}{22\!\cdots\!07}a^{18}+\frac{70\!\cdots\!32}{22\!\cdots\!07}a^{17}-\frac{19\!\cdots\!68}{22\!\cdots\!07}a^{16}-\frac{50\!\cdots\!07}{22\!\cdots\!07}a^{15}+\frac{13\!\cdots\!18}{22\!\cdots\!07}a^{14}+\frac{19\!\cdots\!47}{22\!\cdots\!07}a^{13}+\frac{26\!\cdots\!74}{22\!\cdots\!07}a^{12}-\frac{41\!\cdots\!69}{22\!\cdots\!07}a^{11}-\frac{90\!\cdots\!97}{22\!\cdots\!07}a^{10}+\frac{49\!\cdots\!74}{22\!\cdots\!07}a^{9}+\frac{12\!\cdots\!04}{22\!\cdots\!07}a^{8}-\frac{32\!\cdots\!28}{22\!\cdots\!07}a^{7}-\frac{78\!\cdots\!23}{22\!\cdots\!07}a^{6}+\frac{10\!\cdots\!77}{22\!\cdots\!07}a^{5}+\frac{22\!\cdots\!38}{22\!\cdots\!07}a^{4}-\frac{12\!\cdots\!34}{22\!\cdots\!07}a^{3}-\frac{19\!\cdots\!80}{22\!\cdots\!07}a^{2}+\frac{56\!\cdots\!52}{22\!\cdots\!07}a+\frac{21\!\cdots\!62}{22\!\cdots\!07}$, $\frac{34\!\cdots\!45}{22\!\cdots\!07}a^{21}-\frac{74\!\cdots\!11}{22\!\cdots\!07}a^{20}-\frac{11\!\cdots\!93}{22\!\cdots\!07}a^{19}+\frac{19\!\cdots\!83}{22\!\cdots\!07}a^{18}+\frac{15\!\cdots\!12}{22\!\cdots\!07}a^{17}-\frac{20\!\cdots\!45}{22\!\cdots\!07}a^{16}-\frac{10\!\cdots\!75}{22\!\cdots\!07}a^{15}+\frac{95\!\cdots\!76}{22\!\cdots\!07}a^{14}+\frac{36\!\cdots\!82}{22\!\cdots\!07}a^{13}-\frac{22\!\cdots\!03}{22\!\cdots\!07}a^{12}-\frac{70\!\cdots\!61}{22\!\cdots\!07}a^{11}+\frac{25\!\cdots\!77}{22\!\cdots\!07}a^{10}+\frac{78\!\cdots\!46}{22\!\cdots\!07}a^{9}-\frac{13\!\cdots\!14}{22\!\cdots\!07}a^{8}-\frac{46\!\cdots\!95}{22\!\cdots\!07}a^{7}+\frac{10\!\cdots\!73}{22\!\cdots\!07}a^{6}+\frac{13\!\cdots\!75}{22\!\cdots\!07}a^{5}+\frac{88\!\cdots\!91}{22\!\cdots\!07}a^{4}-\frac{11\!\cdots\!04}{22\!\cdots\!07}a^{3}-\frac{38\!\cdots\!27}{22\!\cdots\!07}a^{2}+\frac{16\!\cdots\!13}{22\!\cdots\!07}a-\frac{38\!\cdots\!24}{22\!\cdots\!07}$, $\frac{52\!\cdots\!10}{22\!\cdots\!07}a^{21}-\frac{56\!\cdots\!07}{22\!\cdots\!07}a^{20}-\frac{11\!\cdots\!98}{22\!\cdots\!07}a^{19}+\frac{20\!\cdots\!71}{22\!\cdots\!07}a^{18}+\frac{73\!\cdots\!00}{22\!\cdots\!07}a^{17}-\frac{29\!\cdots\!41}{22\!\cdots\!07}a^{16}+\frac{23\!\cdots\!69}{22\!\cdots\!07}a^{15}+\frac{21\!\cdots\!89}{22\!\cdots\!07}a^{14}-\frac{47\!\cdots\!76}{22\!\cdots\!07}a^{13}-\frac{84\!\cdots\!07}{22\!\cdots\!07}a^{12}+\frac{22\!\cdots\!60}{22\!\cdots\!07}a^{11}+\frac{18\!\cdots\!31}{22\!\cdots\!07}a^{10}-\frac{50\!\cdots\!41}{22\!\cdots\!07}a^{9}-\frac{23\!\cdots\!17}{22\!\cdots\!07}a^{8}+\frac{53\!\cdots\!19}{22\!\cdots\!07}a^{7}+\frac{15\!\cdots\!66}{22\!\cdots\!07}a^{6}-\frac{25\!\cdots\!92}{22\!\cdots\!07}a^{5}-\frac{46\!\cdots\!91}{22\!\cdots\!07}a^{4}+\frac{47\!\cdots\!10}{22\!\cdots\!07}a^{3}+\frac{48\!\cdots\!66}{22\!\cdots\!07}a^{2}-\frac{40\!\cdots\!83}{22\!\cdots\!07}a-\frac{87\!\cdots\!29}{22\!\cdots\!07}$, $\frac{35\!\cdots\!95}{22\!\cdots\!07}a^{21}-\frac{36\!\cdots\!92}{22\!\cdots\!07}a^{20}-\frac{12\!\cdots\!88}{22\!\cdots\!07}a^{19}+\frac{64\!\cdots\!34}{22\!\cdots\!07}a^{18}+\frac{17\!\cdots\!67}{22\!\cdots\!07}a^{17}-\frac{15\!\cdots\!23}{22\!\cdots\!07}a^{16}-\frac{12\!\cdots\!04}{22\!\cdots\!07}a^{15}-\frac{29\!\cdots\!33}{22\!\cdots\!07}a^{14}+\frac{43\!\cdots\!54}{22\!\cdots\!07}a^{13}+\frac{21\!\cdots\!35}{22\!\cdots\!07}a^{12}-\frac{81\!\cdots\!83}{22\!\cdots\!07}a^{11}-\frac{54\!\cdots\!32}{22\!\cdots\!07}a^{10}+\frac{82\!\cdots\!00}{22\!\cdots\!07}a^{9}+\frac{61\!\cdots\!26}{22\!\cdots\!07}a^{8}-\frac{40\!\cdots\!31}{22\!\cdots\!07}a^{7}-\frac{29\!\cdots\!83}{22\!\cdots\!07}a^{6}+\frac{77\!\cdots\!04}{22\!\cdots\!07}a^{5}+\frac{37\!\cdots\!72}{22\!\cdots\!07}a^{4}+\frac{44\!\cdots\!36}{22\!\cdots\!07}a^{3}+\frac{67\!\cdots\!79}{22\!\cdots\!07}a^{2}-\frac{70\!\cdots\!97}{22\!\cdots\!07}a-\frac{78\!\cdots\!33}{22\!\cdots\!07}$, $\frac{57\!\cdots\!86}{22\!\cdots\!07}a^{21}-\frac{11\!\cdots\!31}{22\!\cdots\!07}a^{20}-\frac{19\!\cdots\!28}{22\!\cdots\!07}a^{19}+\frac{28\!\cdots\!52}{22\!\cdots\!07}a^{18}+\frac{26\!\cdots\!49}{22\!\cdots\!07}a^{17}-\frac{27\!\cdots\!68}{22\!\cdots\!07}a^{16}-\frac{17\!\cdots\!51}{22\!\cdots\!07}a^{15}+\frac{11\!\cdots\!52}{22\!\cdots\!07}a^{14}+\frac{63\!\cdots\!68}{22\!\cdots\!07}a^{13}-\frac{21\!\cdots\!92}{22\!\cdots\!07}a^{12}-\frac{12\!\cdots\!59}{22\!\cdots\!07}a^{11}+\frac{12\!\cdots\!25}{22\!\cdots\!07}a^{10}+\frac{14\!\cdots\!79}{22\!\cdots\!07}a^{9}+\frac{13\!\cdots\!79}{22\!\cdots\!07}a^{8}-\frac{85\!\cdots\!44}{22\!\cdots\!07}a^{7}-\frac{19\!\cdots\!98}{22\!\cdots\!07}a^{6}+\frac{25\!\cdots\!62}{22\!\cdots\!07}a^{5}+\frac{78\!\cdots\!11}{22\!\cdots\!07}a^{4}-\frac{26\!\cdots\!18}{22\!\cdots\!07}a^{3}-\frac{75\!\cdots\!39}{22\!\cdots\!07}a^{2}+\frac{86\!\cdots\!78}{22\!\cdots\!07}a+\frac{10\!\cdots\!83}{22\!\cdots\!07}$, $\frac{72\!\cdots\!99}{22\!\cdots\!07}a^{21}-\frac{30\!\cdots\!30}{22\!\cdots\!07}a^{20}-\frac{20\!\cdots\!68}{22\!\cdots\!07}a^{19}+\frac{88\!\cdots\!40}{22\!\cdots\!07}a^{18}+\frac{20\!\cdots\!10}{22\!\cdots\!07}a^{17}-\frac{98\!\cdots\!08}{22\!\cdots\!07}a^{16}-\frac{87\!\cdots\!46}{22\!\cdots\!07}a^{15}+\frac{52\!\cdots\!30}{22\!\cdots\!07}a^{14}+\frac{11\!\cdots\!74}{22\!\cdots\!07}a^{13}-\frac{14\!\cdots\!39}{22\!\cdots\!07}a^{12}+\frac{18\!\cdots\!60}{22\!\cdots\!07}a^{11}+\frac{20\!\cdots\!50}{22\!\cdots\!07}a^{10}-\frac{58\!\cdots\!93}{22\!\cdots\!07}a^{9}-\frac{15\!\cdots\!00}{22\!\cdots\!07}a^{8}+\frac{50\!\cdots\!88}{22\!\cdots\!07}a^{7}+\frac{58\!\cdots\!92}{22\!\cdots\!07}a^{6}-\frac{14\!\cdots\!42}{22\!\cdots\!07}a^{5}-\frac{82\!\cdots\!56}{22\!\cdots\!07}a^{4}+\frac{15\!\cdots\!37}{22\!\cdots\!07}a^{3}-\frac{40\!\cdots\!04}{22\!\cdots\!07}a^{2}+\frac{20\!\cdots\!18}{22\!\cdots\!07}a+\frac{33\!\cdots\!74}{22\!\cdots\!07}$, $\frac{18\!\cdots\!56}{22\!\cdots\!07}a^{21}-\frac{85\!\cdots\!79}{22\!\cdots\!07}a^{20}-\frac{48\!\cdots\!62}{22\!\cdots\!07}a^{19}+\frac{25\!\cdots\!23}{22\!\cdots\!07}a^{18}+\frac{43\!\cdots\!04}{22\!\cdots\!07}a^{17}-\frac{28\!\cdots\!85}{22\!\cdots\!07}a^{16}-\frac{12\!\cdots\!92}{22\!\cdots\!07}a^{15}+\frac{15\!\cdots\!29}{22\!\cdots\!07}a^{14}-\frac{21\!\cdots\!14}{22\!\cdots\!07}a^{13}-\frac{44\!\cdots\!83}{22\!\cdots\!07}a^{12}+\frac{18\!\cdots\!30}{22\!\cdots\!07}a^{11}+\frac{67\!\cdots\!47}{22\!\cdots\!07}a^{10}-\frac{33\!\cdots\!50}{22\!\cdots\!07}a^{9}-\frac{56\!\cdots\!87}{22\!\cdots\!07}a^{8}+\frac{27\!\cdots\!91}{22\!\cdots\!07}a^{7}+\frac{25\!\cdots\!58}{22\!\cdots\!07}a^{6}-\frac{98\!\cdots\!84}{22\!\cdots\!07}a^{5}-\frac{57\!\cdots\!67}{22\!\cdots\!07}a^{4}+\frac{11\!\cdots\!41}{22\!\cdots\!07}a^{3}+\frac{44\!\cdots\!86}{22\!\cdots\!07}a^{2}-\frac{44\!\cdots\!83}{22\!\cdots\!07}a-\frac{68\!\cdots\!79}{22\!\cdots\!07}$, $\frac{31\!\cdots\!43}{22\!\cdots\!07}a^{21}-\frac{81\!\cdots\!82}{22\!\cdots\!07}a^{20}-\frac{10\!\cdots\!16}{22\!\cdots\!07}a^{19}+\frac{22\!\cdots\!43}{22\!\cdots\!07}a^{18}+\frac{13\!\cdots\!54}{22\!\cdots\!07}a^{17}-\frac{23\!\cdots\!78}{22\!\cdots\!07}a^{16}-\frac{84\!\cdots\!96}{22\!\cdots\!07}a^{15}+\frac{11\!\cdots\!06}{22\!\cdots\!07}a^{14}+\frac{28\!\cdots\!26}{22\!\cdots\!07}a^{13}-\frac{29\!\cdots\!64}{22\!\cdots\!07}a^{12}-\frac{54\!\cdots\!40}{22\!\cdots\!07}a^{11}+\frac{36\!\cdots\!34}{22\!\cdots\!07}a^{10}+\frac{58\!\cdots\!46}{22\!\cdots\!07}a^{9}-\frac{19\!\cdots\!46}{22\!\cdots\!07}a^{8}-\frac{36\!\cdots\!70}{22\!\cdots\!07}a^{7}+\frac{17\!\cdots\!81}{22\!\cdots\!07}a^{6}+\frac{11\!\cdots\!94}{22\!\cdots\!07}a^{5}+\frac{19\!\cdots\!46}{22\!\cdots\!07}a^{4}-\frac{13\!\cdots\!27}{22\!\cdots\!07}a^{3}-\frac{34\!\cdots\!66}{22\!\cdots\!07}a^{2}+\frac{45\!\cdots\!86}{22\!\cdots\!07}a+\frac{10\!\cdots\!03}{22\!\cdots\!07}$, $\frac{94\!\cdots\!70}{22\!\cdots\!07}a^{21}-\frac{27\!\cdots\!78}{22\!\cdots\!07}a^{20}-\frac{30\!\cdots\!03}{22\!\cdots\!07}a^{19}+\frac{78\!\cdots\!14}{22\!\cdots\!07}a^{18}+\frac{36\!\cdots\!87}{22\!\cdots\!07}a^{17}-\frac{84\!\cdots\!11}{22\!\cdots\!07}a^{16}-\frac{20\!\cdots\!84}{22\!\cdots\!07}a^{15}+\frac{43\!\cdots\!06}{22\!\cdots\!07}a^{14}+\frac{58\!\cdots\!80}{22\!\cdots\!07}a^{13}-\frac{11\!\cdots\!53}{22\!\cdots\!07}a^{12}-\frac{79\!\cdots\!98}{22\!\cdots\!07}a^{11}+\frac{16\!\cdots\!86}{22\!\cdots\!07}a^{10}+\frac{42\!\cdots\!40}{22\!\cdots\!07}a^{9}-\frac{13\!\cdots\!67}{22\!\cdots\!07}a^{8}+\frac{69\!\cdots\!39}{22\!\cdots\!07}a^{7}+\frac{62\!\cdots\!96}{22\!\cdots\!07}a^{6}-\frac{13\!\cdots\!83}{22\!\cdots\!07}a^{5}-\frac{15\!\cdots\!36}{22\!\cdots\!07}a^{4}+\frac{34\!\cdots\!33}{22\!\cdots\!07}a^{3}+\frac{19\!\cdots\!24}{22\!\cdots\!07}a^{2}-\frac{19\!\cdots\!17}{22\!\cdots\!07}a-\frac{45\!\cdots\!31}{22\!\cdots\!07}$, $\frac{23\!\cdots\!51}{22\!\cdots\!07}a^{21}-\frac{42\!\cdots\!27}{22\!\cdots\!07}a^{20}-\frac{83\!\cdots\!33}{22\!\cdots\!07}a^{19}+\frac{10\!\cdots\!40}{22\!\cdots\!07}a^{18}+\frac{11\!\cdots\!37}{22\!\cdots\!07}a^{17}-\frac{10\!\cdots\!64}{22\!\cdots\!07}a^{16}-\frac{75\!\cdots\!07}{22\!\cdots\!07}a^{15}+\frac{42\!\cdots\!65}{22\!\cdots\!07}a^{14}+\frac{27\!\cdots\!10}{22\!\cdots\!07}a^{13}-\frac{79\!\cdots\!96}{22\!\cdots\!07}a^{12}-\frac{53\!\cdots\!38}{22\!\cdots\!07}a^{11}+\frac{50\!\cdots\!79}{22\!\cdots\!07}a^{10}+\frac{58\!\cdots\!92}{22\!\cdots\!07}a^{9}+\frac{31\!\cdots\!76}{22\!\cdots\!07}a^{8}-\frac{33\!\cdots\!30}{22\!\cdots\!07}a^{7}-\frac{55\!\cdots\!05}{22\!\cdots\!07}a^{6}+\frac{93\!\cdots\!95}{22\!\cdots\!07}a^{5}+\frac{22\!\cdots\!29}{22\!\cdots\!07}a^{4}-\frac{98\!\cdots\!04}{22\!\cdots\!07}a^{3}-\frac{22\!\cdots\!46}{22\!\cdots\!07}a^{2}+\frac{29\!\cdots\!82}{22\!\cdots\!07}a+\frac{37\!\cdots\!71}{22\!\cdots\!07}$, $\frac{28\!\cdots\!00}{22\!\cdots\!07}a^{21}-\frac{25\!\cdots\!02}{22\!\cdots\!07}a^{20}-\frac{10\!\cdots\!09}{22\!\cdots\!07}a^{19}+\frac{48\!\cdots\!14}{22\!\cdots\!07}a^{18}+\frac{15\!\cdots\!08}{22\!\cdots\!07}a^{17}-\frac{22\!\cdots\!26}{22\!\cdots\!07}a^{16}-\frac{11\!\cdots\!74}{22\!\cdots\!07}a^{15}-\frac{83\!\cdots\!59}{22\!\cdots\!07}a^{14}+\frac{45\!\cdots\!16}{22\!\cdots\!07}a^{13}+\frac{91\!\cdots\!68}{22\!\cdots\!07}a^{12}-\frac{97\!\cdots\!65}{22\!\cdots\!07}a^{11}-\frac{26\!\cdots\!36}{22\!\cdots\!07}a^{10}+\frac{11\!\cdots\!06}{22\!\cdots\!07}a^{9}+\frac{34\!\cdots\!73}{22\!\cdots\!07}a^{8}-\frac{73\!\cdots\!40}{22\!\cdots\!07}a^{7}-\frac{22\!\cdots\!96}{22\!\cdots\!07}a^{6}+\frac{22\!\cdots\!55}{22\!\cdots\!07}a^{5}+\frac{65\!\cdots\!74}{22\!\cdots\!07}a^{4}-\frac{22\!\cdots\!21}{22\!\cdots\!07}a^{3}-\frac{57\!\cdots\!33}{22\!\cdots\!07}a^{2}+\frac{68\!\cdots\!36}{22\!\cdots\!07}a+\frac{95\!\cdots\!88}{22\!\cdots\!07}$, $\frac{97\!\cdots\!42}{22\!\cdots\!07}a^{21}-\frac{57\!\cdots\!69}{22\!\cdots\!07}a^{20}-\frac{23\!\cdots\!09}{22\!\cdots\!07}a^{19}+\frac{17\!\cdots\!86}{22\!\cdots\!07}a^{18}+\frac{15\!\cdots\!96}{22\!\cdots\!07}a^{17}-\frac{20\!\cdots\!96}{22\!\cdots\!07}a^{16}+\frac{17\!\cdots\!34}{22\!\cdots\!07}a^{15}+\frac{11\!\cdots\!61}{22\!\cdots\!07}a^{14}-\frac{55\!\cdots\!43}{22\!\cdots\!07}a^{13}-\frac{34\!\cdots\!56}{22\!\cdots\!07}a^{12}+\frac{21\!\cdots\!78}{22\!\cdots\!07}a^{11}+\frac{54\!\cdots\!17}{22\!\cdots\!07}a^{10}-\frac{34\!\cdots\!04}{22\!\cdots\!07}a^{9}-\frac{49\!\cdots\!13}{22\!\cdots\!07}a^{8}+\frac{26\!\cdots\!58}{22\!\cdots\!07}a^{7}+\frac{23\!\cdots\!95}{22\!\cdots\!07}a^{6}-\frac{94\!\cdots\!02}{22\!\cdots\!07}a^{5}-\frac{54\!\cdots\!54}{22\!\cdots\!07}a^{4}+\frac{12\!\cdots\!82}{22\!\cdots\!07}a^{3}+\frac{40\!\cdots\!85}{22\!\cdots\!07}a^{2}-\frac{67\!\cdots\!54}{22\!\cdots\!07}a-\frac{28\!\cdots\!42}{22\!\cdots\!07}$, $a-1$, $\frac{48\!\cdots\!49}{22\!\cdots\!07}a^{21}-\frac{11\!\cdots\!85}{22\!\cdots\!07}a^{20}-\frac{16\!\cdots\!78}{22\!\cdots\!07}a^{19}+\frac{30\!\cdots\!05}{22\!\cdots\!07}a^{18}+\frac{21\!\cdots\!35}{22\!\cdots\!07}a^{17}-\frac{31\!\cdots\!09}{22\!\cdots\!07}a^{16}-\frac{14\!\cdots\!48}{22\!\cdots\!07}a^{15}+\frac{15\!\cdots\!35}{22\!\cdots\!07}a^{14}+\frac{49\!\cdots\!98}{22\!\cdots\!07}a^{13}-\frac{37\!\cdots\!76}{22\!\cdots\!07}a^{12}-\frac{94\!\cdots\!19}{22\!\cdots\!07}a^{11}+\frac{46\!\cdots\!96}{22\!\cdots\!07}a^{10}+\frac{10\!\cdots\!72}{22\!\cdots\!07}a^{9}-\frac{29\!\cdots\!68}{22\!\cdots\!07}a^{8}-\frac{60\!\cdots\!28}{22\!\cdots\!07}a^{7}+\frac{84\!\cdots\!08}{22\!\cdots\!07}a^{6}+\frac{16\!\cdots\!72}{22\!\cdots\!07}a^{5}-\frac{83\!\cdots\!28}{22\!\cdots\!07}a^{4}-\frac{12\!\cdots\!26}{22\!\cdots\!07}a^{3}+\frac{10\!\cdots\!42}{22\!\cdots\!07}a^{2}-\frac{11\!\cdots\!07}{22\!\cdots\!07}a+\frac{36\!\cdots\!05}{22\!\cdots\!07}$, $\frac{43\!\cdots\!78}{22\!\cdots\!07}a^{21}-\frac{25\!\cdots\!36}{22\!\cdots\!07}a^{20}-\frac{11\!\cdots\!96}{22\!\cdots\!07}a^{19}+\frac{79\!\cdots\!79}{22\!\cdots\!07}a^{18}+\frac{99\!\cdots\!45}{22\!\cdots\!07}a^{17}-\frac{98\!\cdots\!09}{22\!\cdots\!07}a^{16}-\frac{29\!\cdots\!69}{22\!\cdots\!07}a^{15}+\frac{60\!\cdots\!44}{22\!\cdots\!07}a^{14}-\frac{40\!\cdots\!82}{22\!\cdots\!07}a^{13}-\frac{19\!\cdots\!67}{22\!\cdots\!07}a^{12}+\frac{37\!\cdots\!63}{22\!\cdots\!07}a^{11}+\frac{36\!\cdots\!00}{22\!\cdots\!07}a^{10}-\frac{77\!\cdots\!77}{22\!\cdots\!07}a^{9}-\frac{37\!\cdots\!40}{22\!\cdots\!07}a^{8}+\frac{73\!\cdots\!87}{22\!\cdots\!07}a^{7}+\frac{21\!\cdots\!63}{22\!\cdots\!07}a^{6}-\frac{34\!\cdots\!97}{22\!\cdots\!07}a^{5}-\frac{53\!\cdots\!71}{22\!\cdots\!07}a^{4}+\frac{78\!\cdots\!72}{22\!\cdots\!07}a^{3}+\frac{36\!\cdots\!12}{22\!\cdots\!07}a^{2}-\frac{75\!\cdots\!15}{22\!\cdots\!07}a-\frac{12\!\cdots\!93}{22\!\cdots\!07}$, $\frac{19\!\cdots\!03}{22\!\cdots\!07}a^{21}-\frac{31\!\cdots\!70}{22\!\cdots\!07}a^{20}-\frac{70\!\cdots\!43}{22\!\cdots\!07}a^{19}+\frac{75\!\cdots\!28}{22\!\cdots\!07}a^{18}+\frac{97\!\cdots\!00}{22\!\cdots\!07}a^{17}-\frac{65\!\cdots\!92}{22\!\cdots\!07}a^{16}-\frac{66\!\cdots\!42}{22\!\cdots\!07}a^{15}+\frac{23\!\cdots\!87}{22\!\cdots\!07}a^{14}+\frac{24\!\cdots\!53}{22\!\cdots\!07}a^{13}-\frac{20\!\cdots\!52}{22\!\cdots\!07}a^{12}-\frac{49\!\cdots\!59}{22\!\cdots\!07}a^{11}-\frac{59\!\cdots\!56}{22\!\cdots\!07}a^{10}+\frac{55\!\cdots\!73}{22\!\cdots\!07}a^{9}+\frac{15\!\cdots\!96}{22\!\cdots\!07}a^{8}-\frac{31\!\cdots\!20}{22\!\cdots\!07}a^{7}-\frac{14\!\cdots\!00}{22\!\cdots\!07}a^{6}+\frac{74\!\cdots\!14}{22\!\cdots\!07}a^{5}+\frac{51\!\cdots\!93}{22\!\cdots\!07}a^{4}-\frac{56\!\cdots\!65}{22\!\cdots\!07}a^{3}-\frac{52\!\cdots\!53}{22\!\cdots\!07}a^{2}-\frac{44\!\cdots\!29}{22\!\cdots\!07}a+\frac{84\!\cdots\!25}{22\!\cdots\!07}$, $\frac{38\!\cdots\!17}{22\!\cdots\!07}a^{21}-\frac{70\!\cdots\!82}{22\!\cdots\!07}a^{20}-\frac{13\!\cdots\!71}{22\!\cdots\!07}a^{19}+\frac{17\!\cdots\!16}{22\!\cdots\!07}a^{18}+\frac{18\!\cdots\!25}{22\!\cdots\!07}a^{17}-\frac{17\!\cdots\!12}{22\!\cdots\!07}a^{16}-\frac{12\!\cdots\!56}{22\!\cdots\!07}a^{15}+\frac{73\!\cdots\!95}{22\!\cdots\!07}a^{14}+\frac{44\!\cdots\!67}{22\!\cdots\!07}a^{13}-\frac{14\!\cdots\!92}{22\!\cdots\!07}a^{12}-\frac{89\!\cdots\!15}{22\!\cdots\!07}a^{11}+\frac{10\!\cdots\!50}{22\!\cdots\!07}a^{10}+\frac{10\!\cdots\!19}{22\!\cdots\!07}a^{9}+\frac{26\!\cdots\!76}{22\!\cdots\!07}a^{8}-\frac{60\!\cdots\!94}{22\!\cdots\!07}a^{7}-\frac{73\!\cdots\!42}{22\!\cdots\!07}a^{6}+\frac{18\!\cdots\!04}{22\!\cdots\!07}a^{5}+\frac{30\!\cdots\!97}{22\!\cdots\!07}a^{4}-\frac{20\!\cdots\!39}{22\!\cdots\!07}a^{3}-\frac{29\!\cdots\!26}{22\!\cdots\!07}a^{2}+\frac{73\!\cdots\!00}{22\!\cdots\!07}a+\frac{12\!\cdots\!16}{22\!\cdots\!07}$, $\frac{71\!\cdots\!57}{22\!\cdots\!07}a^{21}-\frac{13\!\cdots\!65}{22\!\cdots\!07}a^{20}-\frac{25\!\cdots\!94}{22\!\cdots\!07}a^{19}+\frac{33\!\cdots\!88}{22\!\cdots\!07}a^{18}+\frac{33\!\cdots\!81}{22\!\cdots\!07}a^{17}-\frac{32\!\cdots\!77}{22\!\cdots\!07}a^{16}-\frac{22\!\cdots\!89}{22\!\cdots\!07}a^{15}+\frac{13\!\cdots\!59}{22\!\cdots\!07}a^{14}+\frac{82\!\cdots\!39}{22\!\cdots\!07}a^{13}-\frac{27\!\cdots\!25}{22\!\cdots\!07}a^{12}-\frac{16\!\cdots\!68}{22\!\cdots\!07}a^{11}+\frac{21\!\cdots\!59}{22\!\cdots\!07}a^{10}+\frac{18\!\cdots\!12}{22\!\cdots\!07}a^{9}+\frac{22\!\cdots\!66}{22\!\cdots\!07}a^{8}-\frac{11\!\cdots\!53}{22\!\cdots\!07}a^{7}-\frac{12\!\cdots\!55}{22\!\cdots\!07}a^{6}+\frac{32\!\cdots\!47}{22\!\cdots\!07}a^{5}+\frac{53\!\cdots\!58}{22\!\cdots\!07}a^{4}-\frac{34\!\cdots\!42}{22\!\cdots\!07}a^{3}-\frac{46\!\cdots\!53}{22\!\cdots\!07}a^{2}+\frac{10\!\cdots\!47}{22\!\cdots\!07}a-\frac{20\!\cdots\!98}{22\!\cdots\!07}$, $\frac{58\!\cdots\!69}{22\!\cdots\!07}a^{21}-\frac{80\!\cdots\!75}{22\!\cdots\!07}a^{20}-\frac{12\!\cdots\!86}{22\!\cdots\!07}a^{19}+\frac{29\!\cdots\!31}{22\!\cdots\!07}a^{18}+\frac{79\!\cdots\!35}{22\!\cdots\!07}a^{17}-\frac{41\!\cdots\!46}{22\!\cdots\!07}a^{16}-\frac{10\!\cdots\!63}{22\!\cdots\!07}a^{15}+\frac{30\!\cdots\!38}{22\!\cdots\!07}a^{14}-\frac{19\!\cdots\!89}{22\!\cdots\!07}a^{13}-\frac{11\!\cdots\!19}{22\!\cdots\!07}a^{12}+\frac{96\!\cdots\!26}{22\!\cdots\!07}a^{11}+\frac{25\!\cdots\!92}{22\!\cdots\!07}a^{10}-\frac{22\!\cdots\!30}{22\!\cdots\!07}a^{9}-\frac{29\!\cdots\!29}{22\!\cdots\!07}a^{8}+\frac{23\!\cdots\!17}{22\!\cdots\!07}a^{7}+\frac{18\!\cdots\!68}{22\!\cdots\!07}a^{6}-\frac{88\!\cdots\!19}{22\!\cdots\!07}a^{5}-\frac{52\!\cdots\!14}{22\!\cdots\!07}a^{4}+\frac{41\!\cdots\!55}{22\!\cdots\!07}a^{3}+\frac{47\!\cdots\!41}{22\!\cdots\!07}a^{2}+\frac{28\!\cdots\!65}{22\!\cdots\!07}a-\frac{97\!\cdots\!24}{22\!\cdots\!07}$, $\frac{11\!\cdots\!49}{22\!\cdots\!07}a^{21}-\frac{19\!\cdots\!35}{22\!\cdots\!07}a^{20}-\frac{40\!\cdots\!29}{22\!\cdots\!07}a^{19}+\frac{51\!\cdots\!29}{22\!\cdots\!07}a^{18}+\frac{57\!\cdots\!07}{22\!\cdots\!07}a^{17}-\frac{51\!\cdots\!68}{22\!\cdots\!07}a^{16}-\frac{40\!\cdots\!08}{22\!\cdots\!07}a^{15}+\frac{25\!\cdots\!86}{22\!\cdots\!07}a^{14}+\frac{15\!\cdots\!19}{22\!\cdots\!07}a^{13}-\frac{63\!\cdots\!07}{22\!\cdots\!07}a^{12}-\frac{32\!\cdots\!48}{22\!\cdots\!07}a^{11}+\frac{84\!\cdots\!02}{22\!\cdots\!07}a^{10}+\frac{38\!\cdots\!96}{22\!\cdots\!07}a^{9}-\frac{54\!\cdots\!62}{22\!\cdots\!07}a^{8}-\frac{22\!\cdots\!20}{22\!\cdots\!07}a^{7}+\frac{95\!\cdots\!64}{22\!\cdots\!07}a^{6}+\frac{57\!\cdots\!89}{22\!\cdots\!07}a^{5}+\frac{29\!\cdots\!60}{22\!\cdots\!07}a^{4}-\frac{32\!\cdots\!27}{22\!\cdots\!07}a^{3}-\frac{31\!\cdots\!01}{22\!\cdots\!07}a^{2}-\frac{40\!\cdots\!24}{22\!\cdots\!07}a-\frac{24\!\cdots\!84}{22\!\cdots\!07}$, $\frac{19\!\cdots\!17}{22\!\cdots\!07}a^{21}-\frac{54\!\cdots\!10}{22\!\cdots\!07}a^{20}-\frac{63\!\cdots\!82}{22\!\cdots\!07}a^{19}+\frac{15\!\cdots\!69}{22\!\cdots\!07}a^{18}+\frac{83\!\cdots\!89}{22\!\cdots\!07}a^{17}-\frac{17\!\cdots\!70}{22\!\cdots\!07}a^{16}-\frac{55\!\cdots\!37}{22\!\cdots\!07}a^{15}+\frac{95\!\cdots\!28}{22\!\cdots\!07}a^{14}+\frac{20\!\cdots\!05}{22\!\cdots\!07}a^{13}-\frac{26\!\cdots\!89}{22\!\cdots\!07}a^{12}-\frac{45\!\cdots\!36}{22\!\cdots\!07}a^{11}+\frac{40\!\cdots\!49}{22\!\cdots\!07}a^{10}+\frac{60\!\cdots\!99}{22\!\cdots\!07}a^{9}-\frac{30\!\cdots\!34}{22\!\cdots\!07}a^{8}-\frac{47\!\cdots\!05}{22\!\cdots\!07}a^{7}+\frac{99\!\cdots\!95}{22\!\cdots\!07}a^{6}+\frac{19\!\cdots\!54}{22\!\cdots\!07}a^{5}-\frac{36\!\cdots\!38}{22\!\cdots\!07}a^{4}-\frac{37\!\cdots\!30}{22\!\cdots\!07}a^{3}-\frac{19\!\cdots\!41}{22\!\cdots\!07}a^{2}+\frac{23\!\cdots\!82}{22\!\cdots\!07}a+\frac{11\!\cdots\!20}{22\!\cdots\!07}$
| 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: | \( 25661717777.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 25661717777.2 \cdot 1}{2\cdot\sqrt{115836063662590130514138078564453125}}\cr\approx \mathstrut & 0.158122523924 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*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^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*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^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*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^22 - 3*x^21 - 33*x^20 + 88*x^19 + 421*x^18 - 1001*x^17 - 2683*x^16 + 5647*x^15 + 9370*x^14 - 17186*x^13 - 18923*x^12 + 29509*x^11 + 23007*x^10 - 28981*x^9 - 17000*x^8 + 15763*x^7 + 7336*x^6 - 4249*x^5 - 1613*x^4 + 425*x^3 + 118*x^2 - 13*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
$C_2\times S_{11}$ (as 22T47):
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 non-solvable group of order 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ are not computed |
| Character table for $C_2\times S_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.11.48706494267293.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 sibling: | 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | $22$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.6.0.1}{6} }$ | $22$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.14.7.1 | $x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(421\)
| $\Q_{421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(115692385433\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |