Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 1)
gp: K = bnfinit(y^22 + 44*y^18 - 22*y^17 + 55*y^16 + 121*y^15 + 187*y^14 + 11*y^13 + 594*y^12 + 328*y^11 + 836*y^10 + 770*y^9 + 814*y^8 + 528*y^7 + 374*y^6 + 143*y^5 + 11*y^4 - 33*y^3 - 11*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 1)
\( x^{22} + 44 x^{18} - 22 x^{17} + 55 x^{16} + 121 x^{15} + 187 x^{14} + 11 x^{13} + 594 x^{12} + 328 x^{11} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-13109994191499930367061460371\)
\(\medspace = -\,11^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.97\) | 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: | $11^{139/110}\approx 20.6986330715433$ | ||
| Ramified primes: |
\(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{11\!\cdots\!68}a^{21}+\frac{16\!\cdots\!13}{11\!\cdots\!68}a^{20}+\frac{24\!\cdots\!73}{11\!\cdots\!68}a^{19}-\frac{89\!\cdots\!39}{11\!\cdots\!68}a^{18}-\frac{11\!\cdots\!81}{37\!\cdots\!56}a^{17}-\frac{27\!\cdots\!09}{11\!\cdots\!68}a^{16}-\frac{41\!\cdots\!89}{18\!\cdots\!78}a^{15}+\frac{19\!\cdots\!79}{11\!\cdots\!68}a^{14}+\frac{38\!\cdots\!05}{56\!\cdots\!34}a^{13}-\frac{35\!\cdots\!31}{11\!\cdots\!68}a^{12}+\frac{19\!\cdots\!35}{11\!\cdots\!68}a^{11}-\frac{28\!\cdots\!89}{11\!\cdots\!68}a^{10}-\frac{30\!\cdots\!85}{11\!\cdots\!68}a^{9}-\frac{46\!\cdots\!05}{37\!\cdots\!56}a^{8}-\frac{21\!\cdots\!33}{11\!\cdots\!68}a^{7}-\frac{17\!\cdots\!65}{11\!\cdots\!68}a^{6}+\frac{11\!\cdots\!23}{37\!\cdots\!56}a^{5}+\frac{62\!\cdots\!19}{28\!\cdots\!67}a^{4}-\frac{48\!\cdots\!85}{11\!\cdots\!68}a^{3}-\frac{18\!\cdots\!79}{56\!\cdots\!34}a^{2}-\frac{11\!\cdots\!13}{11\!\cdots\!68}a-\frac{24\!\cdots\!29}{11\!\cdots\!68}$
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: |
$\frac{50\!\cdots\!63}{37\!\cdots\!56}a^{21}-\frac{12\!\cdots\!77}{37\!\cdots\!56}a^{20}-\frac{66\!\cdots\!81}{37\!\cdots\!56}a^{19}+\frac{49\!\cdots\!79}{37\!\cdots\!56}a^{18}+\frac{22\!\cdots\!51}{37\!\cdots\!56}a^{17}-\frac{16\!\cdots\!15}{37\!\cdots\!56}a^{16}+\frac{13\!\cdots\!67}{18\!\cdots\!78}a^{15}+\frac{57\!\cdots\!17}{37\!\cdots\!56}a^{14}+\frac{36\!\cdots\!53}{18\!\cdots\!78}a^{13}-\frac{22\!\cdots\!25}{37\!\cdots\!56}a^{12}+\frac{29\!\cdots\!61}{37\!\cdots\!56}a^{11}+\frac{96\!\cdots\!89}{37\!\cdots\!56}a^{10}+\frac{33\!\cdots\!61}{37\!\cdots\!56}a^{9}+\frac{29\!\cdots\!83}{37\!\cdots\!56}a^{8}+\frac{26\!\cdots\!61}{37\!\cdots\!56}a^{7}+\frac{14\!\cdots\!93}{37\!\cdots\!56}a^{6}+\frac{89\!\cdots\!83}{37\!\cdots\!56}a^{5}+\frac{34\!\cdots\!87}{94\!\cdots\!89}a^{4}-\frac{25\!\cdots\!07}{37\!\cdots\!56}a^{3}-\frac{92\!\cdots\!69}{18\!\cdots\!78}a^{2}-\frac{12\!\cdots\!83}{37\!\cdots\!56}a+\frac{18\!\cdots\!41}{37\!\cdots\!56}$, $\frac{35\!\cdots\!17}{37\!\cdots\!56}a^{21}+\frac{17\!\cdots\!73}{37\!\cdots\!56}a^{20}-\frac{25\!\cdots\!11}{37\!\cdots\!56}a^{19}+\frac{11\!\cdots\!05}{37\!\cdots\!56}a^{18}+\frac{15\!\cdots\!93}{37\!\cdots\!56}a^{17}-\frac{48\!\cdots\!17}{37\!\cdots\!56}a^{16}+\frac{77\!\cdots\!57}{18\!\cdots\!78}a^{15}+\frac{53\!\cdots\!95}{37\!\cdots\!56}a^{14}+\frac{44\!\cdots\!97}{18\!\cdots\!78}a^{13}+\frac{36\!\cdots\!17}{37\!\cdots\!56}a^{12}+\frac{21\!\cdots\!51}{37\!\cdots\!56}a^{11}+\frac{22\!\cdots\!47}{37\!\cdots\!56}a^{10}+\frac{35\!\cdots\!59}{37\!\cdots\!56}a^{9}+\frac{42\!\cdots\!85}{37\!\cdots\!56}a^{8}+\frac{43\!\cdots\!15}{37\!\cdots\!56}a^{7}+\frac{33\!\cdots\!87}{37\!\cdots\!56}a^{6}+\frac{24\!\cdots\!09}{37\!\cdots\!56}a^{5}+\frac{32\!\cdots\!91}{94\!\cdots\!89}a^{4}+\frac{41\!\cdots\!19}{37\!\cdots\!56}a^{3}-\frac{13\!\cdots\!41}{18\!\cdots\!78}a^{2}-\frac{48\!\cdots\!25}{37\!\cdots\!56}a-\frac{13\!\cdots\!97}{37\!\cdots\!56}$, $\frac{25\!\cdots\!63}{37\!\cdots\!56}a^{21}+\frac{14\!\cdots\!95}{37\!\cdots\!56}a^{20}+\frac{96\!\cdots\!87}{37\!\cdots\!56}a^{19}+\frac{21\!\cdots\!75}{37\!\cdots\!56}a^{18}+\frac{11\!\cdots\!51}{37\!\cdots\!56}a^{17}+\frac{69\!\cdots\!01}{37\!\cdots\!56}a^{16}+\frac{55\!\cdots\!29}{18\!\cdots\!78}a^{15}+\frac{38\!\cdots\!29}{37\!\cdots\!56}a^{14}+\frac{32\!\cdots\!77}{18\!\cdots\!78}a^{13}+\frac{31\!\cdots\!79}{37\!\cdots\!56}a^{12}+\frac{15\!\cdots\!25}{37\!\cdots\!56}a^{11}+\frac{17\!\cdots\!41}{37\!\cdots\!56}a^{10}+\frac{26\!\cdots\!37}{37\!\cdots\!56}a^{9}+\frac{32\!\cdots\!23}{37\!\cdots\!56}a^{8}+\frac{32\!\cdots\!05}{37\!\cdots\!56}a^{7}+\frac{27\!\cdots\!41}{37\!\cdots\!56}a^{6}+\frac{19\!\cdots\!07}{37\!\cdots\!56}a^{5}+\frac{27\!\cdots\!58}{94\!\cdots\!89}a^{4}+\frac{38\!\cdots\!45}{37\!\cdots\!56}a^{3}+\frac{86\!\cdots\!37}{18\!\cdots\!78}a^{2}-\frac{33\!\cdots\!95}{37\!\cdots\!56}a-\frac{15\!\cdots\!47}{37\!\cdots\!56}$, $a$, $\frac{57\!\cdots\!60}{94\!\cdots\!89}a^{21}+\frac{27\!\cdots\!48}{94\!\cdots\!89}a^{20}+\frac{11\!\cdots\!92}{94\!\cdots\!89}a^{19}-\frac{34\!\cdots\!79}{94\!\cdots\!89}a^{18}+\frac{25\!\cdots\!66}{94\!\cdots\!89}a^{17}-\frac{50\!\cdots\!87}{94\!\cdots\!89}a^{16}+\frac{30\!\cdots\!52}{94\!\cdots\!89}a^{15}+\frac{80\!\cdots\!55}{94\!\cdots\!89}a^{14}+\frac{15\!\cdots\!19}{94\!\cdots\!89}a^{13}+\frac{68\!\cdots\!64}{94\!\cdots\!89}a^{12}+\frac{36\!\cdots\!69}{94\!\cdots\!89}a^{11}+\frac{35\!\cdots\!35}{94\!\cdots\!89}a^{10}+\frac{64\!\cdots\!58}{94\!\cdots\!89}a^{9}+\frac{69\!\cdots\!16}{94\!\cdots\!89}a^{8}+\frac{80\!\cdots\!61}{94\!\cdots\!89}a^{7}+\frac{59\!\cdots\!27}{94\!\cdots\!89}a^{6}+\frac{46\!\cdots\!37}{94\!\cdots\!89}a^{5}+\frac{24\!\cdots\!16}{94\!\cdots\!89}a^{4}+\frac{99\!\cdots\!65}{94\!\cdots\!89}a^{3}+\frac{39\!\cdots\!32}{94\!\cdots\!89}a^{2}-\frac{75\!\cdots\!46}{94\!\cdots\!89}a-\frac{35\!\cdots\!51}{94\!\cdots\!89}$, $\frac{22\!\cdots\!05}{56\!\cdots\!34}a^{21}-\frac{51\!\cdots\!67}{56\!\cdots\!34}a^{20}+\frac{10\!\cdots\!15}{56\!\cdots\!34}a^{19}-\frac{80\!\cdots\!67}{56\!\cdots\!34}a^{18}+\frac{32\!\cdots\!13}{18\!\cdots\!78}a^{17}-\frac{27\!\cdots\!25}{56\!\cdots\!34}a^{16}+\frac{39\!\cdots\!37}{94\!\cdots\!89}a^{15}-\frac{16\!\cdots\!03}{56\!\cdots\!34}a^{14}-\frac{10\!\cdots\!90}{28\!\cdots\!67}a^{13}-\frac{90\!\cdots\!49}{56\!\cdots\!34}a^{12}+\frac{12\!\cdots\!97}{56\!\cdots\!34}a^{11}-\frac{23\!\cdots\!83}{56\!\cdots\!34}a^{10}+\frac{20\!\cdots\!51}{56\!\cdots\!34}a^{9}-\frac{84\!\cdots\!73}{18\!\cdots\!78}a^{8}-\frac{22\!\cdots\!05}{56\!\cdots\!34}a^{7}-\frac{29\!\cdots\!01}{56\!\cdots\!34}a^{6}-\frac{70\!\cdots\!19}{18\!\cdots\!78}a^{5}-\frac{84\!\cdots\!99}{28\!\cdots\!67}a^{4}-\frac{89\!\cdots\!01}{56\!\cdots\!34}a^{3}-\frac{11\!\cdots\!41}{28\!\cdots\!67}a^{2}+\frac{53\!\cdots\!65}{56\!\cdots\!34}a+\frac{35\!\cdots\!93}{56\!\cdots\!34}$, $\frac{34\!\cdots\!09}{56\!\cdots\!34}a^{21}+\frac{25\!\cdots\!81}{56\!\cdots\!34}a^{20}+\frac{73\!\cdots\!21}{56\!\cdots\!34}a^{19}-\frac{10\!\cdots\!27}{56\!\cdots\!34}a^{18}+\frac{50\!\cdots\!63}{18\!\cdots\!78}a^{17}+\frac{37\!\cdots\!81}{56\!\cdots\!34}a^{16}+\frac{27\!\cdots\!87}{94\!\cdots\!89}a^{15}+\frac{53\!\cdots\!35}{56\!\cdots\!34}a^{14}+\frac{49\!\cdots\!52}{28\!\cdots\!67}a^{13}+\frac{59\!\cdots\!27}{56\!\cdots\!34}a^{12}+\frac{21\!\cdots\!91}{56\!\cdots\!34}a^{11}+\frac{26\!\cdots\!37}{56\!\cdots\!34}a^{10}+\frac{41\!\cdots\!99}{56\!\cdots\!34}a^{9}+\frac{16\!\cdots\!17}{18\!\cdots\!78}a^{8}+\frac{54\!\cdots\!91}{56\!\cdots\!34}a^{7}+\frac{43\!\cdots\!49}{56\!\cdots\!34}a^{6}+\frac{10\!\cdots\!09}{18\!\cdots\!78}a^{5}+\frac{92\!\cdots\!85}{28\!\cdots\!67}a^{4}+\frac{73\!\cdots\!51}{56\!\cdots\!34}a^{3}+\frac{31\!\cdots\!68}{28\!\cdots\!67}a^{2}-\frac{68\!\cdots\!59}{56\!\cdots\!34}a-\frac{26\!\cdots\!19}{56\!\cdots\!34}$, $\frac{22\!\cdots\!61}{11\!\cdots\!68}a^{21}-\frac{87\!\cdots\!47}{11\!\cdots\!68}a^{20}+\frac{15\!\cdots\!21}{11\!\cdots\!68}a^{19}+\frac{11\!\cdots\!41}{11\!\cdots\!68}a^{18}+\frac{33\!\cdots\!75}{37\!\cdots\!56}a^{17}-\frac{88\!\cdots\!89}{11\!\cdots\!68}a^{16}+\frac{25\!\cdots\!27}{18\!\cdots\!78}a^{15}+\frac{23\!\cdots\!75}{11\!\cdots\!68}a^{14}+\frac{16\!\cdots\!05}{56\!\cdots\!34}a^{13}-\frac{11\!\cdots\!07}{11\!\cdots\!68}a^{12}+\frac{13\!\cdots\!63}{11\!\cdots\!68}a^{11}+\frac{24\!\cdots\!15}{11\!\cdots\!68}a^{10}+\frac{17\!\cdots\!35}{11\!\cdots\!68}a^{9}+\frac{38\!\cdots\!71}{37\!\cdots\!56}a^{8}+\frac{12\!\cdots\!47}{11\!\cdots\!68}a^{7}+\frac{70\!\cdots\!47}{11\!\cdots\!68}a^{6}+\frac{17\!\cdots\!79}{37\!\cdots\!56}a^{5}+\frac{31\!\cdots\!02}{28\!\cdots\!67}a^{4}-\frac{33\!\cdots\!01}{11\!\cdots\!68}a^{3}-\frac{23\!\cdots\!95}{56\!\cdots\!34}a^{2}+\frac{21\!\cdots\!15}{11\!\cdots\!68}a+\frac{22\!\cdots\!71}{11\!\cdots\!68}$, $\frac{67\!\cdots\!37}{37\!\cdots\!56}a^{21}+\frac{50\!\cdots\!69}{37\!\cdots\!56}a^{20}-\frac{13\!\cdots\!87}{37\!\cdots\!56}a^{19}+\frac{95\!\cdots\!93}{37\!\cdots\!56}a^{18}+\frac{29\!\cdots\!13}{37\!\cdots\!56}a^{17}-\frac{12\!\cdots\!33}{37\!\cdots\!56}a^{16}+\frac{14\!\cdots\!93}{18\!\cdots\!78}a^{15}+\frac{91\!\cdots\!27}{37\!\cdots\!56}a^{14}+\frac{60\!\cdots\!67}{18\!\cdots\!78}a^{13}+\frac{61\!\cdots\!89}{37\!\cdots\!56}a^{12}+\frac{38\!\cdots\!35}{37\!\cdots\!56}a^{11}+\frac{26\!\cdots\!47}{37\!\cdots\!56}a^{10}+\frac{49\!\cdots\!43}{37\!\cdots\!56}a^{9}+\frac{57\!\cdots\!05}{37\!\cdots\!56}a^{8}+\frac{48\!\cdots\!31}{37\!\cdots\!56}a^{7}+\frac{35\!\cdots\!35}{37\!\cdots\!56}a^{6}+\frac{21\!\cdots\!93}{37\!\cdots\!56}a^{5}+\frac{23\!\cdots\!04}{94\!\cdots\!89}a^{4}-\frac{88\!\cdots\!61}{37\!\cdots\!56}a^{3}-\frac{10\!\cdots\!15}{18\!\cdots\!78}a^{2}-\frac{61\!\cdots\!01}{37\!\cdots\!56}a+\frac{19\!\cdots\!83}{37\!\cdots\!56}$, $\frac{72\!\cdots\!83}{37\!\cdots\!56}a^{21}+\frac{92\!\cdots\!11}{37\!\cdots\!56}a^{20}+\frac{87\!\cdots\!63}{37\!\cdots\!56}a^{19}+\frac{61\!\cdots\!27}{37\!\cdots\!56}a^{18}+\frac{31\!\cdots\!03}{37\!\cdots\!56}a^{17}-\frac{11\!\cdots\!91}{37\!\cdots\!56}a^{16}+\frac{20\!\cdots\!99}{18\!\cdots\!78}a^{15}+\frac{90\!\cdots\!57}{37\!\cdots\!56}a^{14}+\frac{75\!\cdots\!73}{18\!\cdots\!78}a^{13}+\frac{36\!\cdots\!71}{37\!\cdots\!56}a^{12}+\frac{44\!\cdots\!69}{37\!\cdots\!56}a^{11}+\frac{29\!\cdots\!01}{37\!\cdots\!56}a^{10}+\frac{68\!\cdots\!37}{37\!\cdots\!56}a^{9}+\frac{66\!\cdots\!95}{37\!\cdots\!56}a^{8}+\frac{73\!\cdots\!01}{37\!\cdots\!56}a^{7}+\frac{53\!\cdots\!97}{37\!\cdots\!56}a^{6}+\frac{39\!\cdots\!07}{37\!\cdots\!56}a^{5}+\frac{48\!\cdots\!74}{94\!\cdots\!89}a^{4}+\frac{61\!\cdots\!37}{37\!\cdots\!56}a^{3}-\frac{15\!\cdots\!55}{18\!\cdots\!78}a^{2}-\frac{62\!\cdots\!59}{37\!\cdots\!56}a-\frac{18\!\cdots\!51}{37\!\cdots\!56}$
| 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: | \( 302171.954912 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 302171.954912 \cdot 1}{2\cdot\sqrt{13109994191499930367061460371}}\cr\approx \mathstrut & 0.795062849568 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 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 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 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 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 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 + 44*x^18 - 22*x^17 + 55*x^16 + 121*x^15 + 187*x^14 + 11*x^13 + 594*x^12 + 328*x^11 + 836*x^10 + 770*x^9 + 814*x^8 + 528*x^7 + 374*x^6 + 143*x^5 + 11*x^4 - 33*x^3 - 11*x^2 + 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 110 |
| The 11 conjugacy class representatives for $F_{11}$ |
| Character table for $F_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 11.1.34522712143931.1 |
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 11 sibling: | data not computed |
| Minimal sibling: | 11.1.34522712143931.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{11}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| Deg $22$ | $22$ | $1$ | $27$ |