Properties

Label 22.20.158...999.1
Degree $22$
Signature $[20, 1]$
Discriminant $-1.589\times 10^{34}$
Root discriminant \(35.86\)
Ramified primes $79,1451,18521,71823490019$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.(C_2\times S_{11})$ (as 22T53)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*x - 1)
 
gp: K = bnfinit(y^22 - 11*y^21 + 41*y^20 - 25*y^19 - 223*y^18 + 525*y^17 + 92*y^16 - 1654*y^15 + 1480*y^14 + 1772*y^13 - 3759*y^12 + 506*y^11 + 3901*y^10 - 2800*y^9 - 1867*y^8 + 2565*y^7 + 335*y^6 - 1066*y^5 - 4*y^4 + 202*y^3 + 3*y^2 - 14*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*x - 1)
 

\( x^{22} - 11 x^{21} + 41 x^{20} - 25 x^{19} - 223 x^{18} + 525 x^{17} + 92 x^{16} - 1654 x^{15} + 1480 x^{14} + 1772 x^{13} - 3759 x^{12} + 506 x^{11} + 3901 x^{10} - 2800 x^{9} - 1867 x^{8} + \cdots - 1 \) Copy content Toggle raw display

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:  $[20, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-15891297445653813609959315666834999\) \(\medspace = -\,79\cdot 1451^{2}\cdot 18521\cdot 71823490019^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(35.86\)
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:  $79^{1/2}1451^{1/2}18521^{1/2}71823490019^{1/2}\approx 12348457743.510412$
Ramified primes:   \(79\), \(1451\), \(18521\), \(71823490019\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1463159}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7}a^{18}-\frac{2}{7}a^{17}+\frac{1}{7}a^{15}-\frac{2}{7}a^{14}-\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{9}+\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{2}+\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{19}+\frac{3}{7}a^{17}+\frac{1}{7}a^{16}+\frac{3}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}-\frac{3}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{14}+\frac{3}{7}a^{13}-\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{3}{7}a^{10}+\frac{3}{7}a^{9}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{21}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}-\frac{1}{7}a^{13}+\frac{3}{7}a^{12}+\frac{3}{7}a^{11}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$ Copy content Toggle raw display

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 $2$) Copy content Toggle raw display
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:   $a^{20}-10a^{19}+32a^{18}-3a^{17}-194a^{16}+328a^{15}+226a^{14}-1100a^{13}+606a^{12}+1278a^{11}-1875a^{10}-91a^{9}+1935a^{8}-956a^{7}-888a^{6}+721a^{5}+168a^{4}-177a^{3}-13a^{2}+12a+2$, $2a^{20}-20a^{19}+\frac{444}{7}a^{18}-\frac{6}{7}a^{17}-401a^{16}+\frac{4504}{7}a^{15}+\frac{3879}{7}a^{14}-2287a^{13}+\frac{6864}{7}a^{12}+\frac{20917}{7}a^{11}-3680a^{10}-\frac{6309}{7}a^{9}+\frac{30372}{7}a^{8}-\frac{10061}{7}a^{7}-\frac{17617}{7}a^{6}+\frac{9876}{7}a^{5}+\frac{5342}{7}a^{4}-430a^{3}-\frac{754}{7}a^{2}+\frac{261}{7}a+\frac{27}{7}$, $\frac{16}{7}a^{20}-\frac{160}{7}a^{19}+\frac{498}{7}a^{18}+\frac{78}{7}a^{17}-\frac{3436}{7}a^{16}+\frac{5048}{7}a^{15}+\frac{5891}{7}a^{14}-\frac{20029}{7}a^{13}+809a^{12}+4285a^{11}-\frac{31350}{7}a^{10}-\frac{14176}{7}a^{9}+\frac{42650}{7}a^{8}-1431a^{7}-\frac{27611}{7}a^{6}+\frac{13473}{7}a^{5}+\frac{8839}{7}a^{4}-660a^{3}-\frac{1187}{7}a^{2}+\frac{435}{7}a+\frac{50}{7}$, $\frac{5}{7}a^{20}-\frac{50}{7}a^{19}+\frac{157}{7}a^{18}+\frac{12}{7}a^{17}-\frac{1044}{7}a^{16}+\frac{1620}{7}a^{15}+\frac{1570}{7}a^{14}-\frac{6088}{7}a^{13}+\frac{2516}{7}a^{12}+\frac{8252}{7}a^{11}-\frac{10391}{7}a^{10}-\frac{2195}{7}a^{9}+\frac{12483}{7}a^{8}-\frac{5136}{7}a^{7}-\frac{6551}{7}a^{6}+\frac{5008}{7}a^{5}+\frac{1170}{7}a^{4}-223a^{3}+\frac{93}{7}a^{2}+\frac{130}{7}a-2$, $3a^{20}-30a^{19}+\frac{668}{7}a^{18}-\frac{27}{7}a^{17}-595a^{16}+\frac{6800}{7}a^{15}+\frac{5461}{7}a^{14}-3387a^{13}+\frac{11106}{7}a^{12}+\frac{29863}{7}a^{11}-5555a^{10}-\frac{6946}{7}a^{9}+\frac{43917}{7}a^{8}-\frac{16753}{7}a^{7}-\frac{23833}{7}a^{6}+\frac{14923}{7}a^{5}+\frac{6518}{7}a^{4}-607a^{3}-\frac{838}{7}a^{2}+\frac{338}{7}a+\frac{34}{7}$, $a$, $a-1$, $2a^{20}-20a^{19}+\frac{447}{7}a^{18}-\frac{33}{7}a^{17}-391a^{16}+\frac{4556}{7}a^{15}+\frac{3376}{7}a^{14}-2220a^{13}+\frac{7911}{7}a^{12}+\frac{18860}{7}a^{11}-3717a^{10}-\frac{3056}{7}a^{9}+\frac{28341}{7}a^{8}-\frac{12310}{7}a^{7}-\frac{14388}{7}a^{6}+\frac{10351}{7}a^{5}+\frac{3299}{7}a^{4}-416a^{3}-\frac{255}{7}a^{2}+\frac{235}{7}a+\frac{12}{7}$, $\frac{2}{7}a^{20}-\frac{20}{7}a^{19}+\frac{54}{7}a^{18}+12a^{17}-\frac{629}{7}a^{16}+\frac{544}{7}a^{15}+\frac{2012}{7}a^{14}-\frac{4020}{7}a^{13}-\frac{1201}{7}a^{12}+\frac{9078}{7}a^{11}-\frac{5590}{7}a^{10}-\frac{7867}{7}a^{9}+1754a^{8}+\frac{44}{7}a^{7}-\frac{9994}{7}a^{6}+\frac{3597}{7}a^{5}+\frac{3497}{7}a^{4}-230a^{3}-\frac{433}{7}a^{2}+\frac{174}{7}a+\frac{23}{7}$, $\frac{10}{7}a^{21}-\frac{113}{7}a^{20}+\frac{450}{7}a^{19}-\frac{449}{7}a^{18}-\frac{1875}{7}a^{17}+\frac{5735}{7}a^{16}-290a^{15}-\frac{13561}{7}a^{14}+\frac{19928}{7}a^{13}+\frac{4483}{7}a^{12}-4884a^{11}+3273a^{10}+2829a^{9}-\frac{33617}{7}a^{8}+\frac{3284}{7}a^{7}+\frac{18596}{7}a^{6}-\frac{7780}{7}a^{5}-\frac{4293}{7}a^{4}+\frac{2612}{7}a^{3}+\frac{330}{7}a^{2}-35a-\frac{2}{7}$, $\frac{13}{7}a^{21}-\frac{139}{7}a^{20}+71a^{19}-\frac{246}{7}a^{18}-\frac{2707}{7}a^{17}+838a^{16}+\frac{1506}{7}a^{15}-\frac{17944}{7}a^{14}+\frac{14956}{7}a^{13}+\frac{17910}{7}a^{12}-\frac{36916}{7}a^{11}+894a^{10}+\frac{35204}{7}a^{9}-3750a^{8}-\frac{13903}{7}a^{7}+2899a^{6}+68a^{5}-\frac{6529}{7}a^{4}+\frac{1046}{7}a^{3}+\frac{876}{7}a^{2}-\frac{163}{7}a-\frac{50}{7}$, $\frac{10}{7}a^{20}-\frac{100}{7}a^{19}+\frac{317}{7}a^{18}-\frac{3}{7}a^{17}-\frac{2011}{7}a^{16}+\frac{3236}{7}a^{15}+\frac{2749}{7}a^{14}-\frac{11511}{7}a^{13}+\frac{5169}{7}a^{12}+\frac{14811}{7}a^{11}-\frac{18927}{7}a^{10}-\frac{3699}{7}a^{9}+\frac{21689}{7}a^{8}-\frac{8181}{7}a^{7}-\frac{11756}{7}a^{6}+\frac{7334}{7}a^{5}+\frac{3195}{7}a^{4}-295a^{3}-\frac{449}{7}a^{2}+\frac{192}{7}a+\frac{27}{7}$, $\frac{6}{7}a^{21}-\frac{74}{7}a^{20}+\frac{330}{7}a^{19}-\frac{439}{7}a^{18}-\frac{1254}{7}a^{17}+\frac{4901}{7}a^{16}-\frac{2866}{7}a^{15}-\frac{11630}{7}a^{14}+\frac{20596}{7}a^{13}+\frac{2585}{7}a^{12}-\frac{36549}{7}a^{11}+\frac{26974}{7}a^{10}+\frac{22716}{7}a^{9}-\frac{42648}{7}a^{8}+722a^{7}+\frac{27317}{7}a^{6}-\frac{12325}{7}a^{5}-\frac{7795}{7}a^{4}+\frac{4833}{7}a^{3}+\frac{829}{7}a^{2}-\frac{548}{7}a-\frac{38}{7}$, $\frac{18}{7}a^{21}-\frac{194}{7}a^{20}+101a^{19}-\frac{417}{7}a^{18}-\frac{3695}{7}a^{17}+1221a^{16}+\frac{1101}{7}a^{15}-\frac{25142}{7}a^{14}+\frac{23788}{7}a^{13}+3229a^{12}-\frac{54504}{7}a^{11}+\frac{14375}{7}a^{10}+\frac{48471}{7}a^{9}-6011a^{8}-\frac{15949}{7}a^{7}+\frac{30797}{7}a^{6}-\frac{1511}{7}a^{5}-\frac{9535}{7}a^{4}+256a^{3}+\frac{1079}{7}a^{2}-\frac{221}{7}a-\frac{20}{7}$, $\frac{4}{7}a^{21}-\frac{31}{7}a^{20}+\frac{29}{7}a^{19}+\frac{369}{7}a^{18}-161a^{17}-\frac{251}{7}a^{16}+\frac{5394}{7}a^{15}-\frac{5813}{7}a^{14}-1083a^{13}+\frac{19221}{7}a^{12}-\frac{4953}{7}a^{11}-3187a^{10}+\frac{23099}{7}a^{9}+\frac{6497}{7}a^{8}-\frac{22273}{7}a^{7}+\frac{5660}{7}a^{6}+\frac{8492}{7}a^{5}-\frac{4135}{7}a^{4}-151a^{3}+114a^{2}+3a-\frac{52}{7}$, $\frac{8}{7}a^{21}-\frac{81}{7}a^{20}+\frac{263}{7}a^{19}-4a^{18}-\frac{1626}{7}a^{17}+\frac{2776}{7}a^{16}+\frac{2014}{7}a^{15}-\frac{9581}{7}a^{14}+\frac{4883}{7}a^{13}+\frac{11863}{7}a^{12}-\frac{16062}{7}a^{11}-325a^{10}+\frac{17568}{7}a^{9}-\frac{7083}{7}a^{8}-\frac{9327}{7}a^{7}+\frac{5636}{7}a^{6}+\frac{2920}{7}a^{5}-\frac{1156}{7}a^{4}-\frac{725}{7}a^{3}-15a^{2}+\frac{102}{7}a+\frac{40}{7}$, $\frac{6}{7}a^{21}-\frac{53}{7}a^{20}+\frac{129}{7}a^{19}+20a^{18}-\frac{985}{7}a^{17}+\frac{584}{7}a^{16}+\frac{2578}{7}a^{15}-446a^{14}-491a^{13}+\frac{6618}{7}a^{12}+\frac{2423}{7}a^{11}-\frac{10104}{7}a^{10}+\frac{1266}{7}a^{9}+\frac{11950}{7}a^{8}-870a^{7}-\frac{8779}{7}a^{6}+\frac{6365}{7}a^{5}+\frac{3163}{7}a^{4}-\frac{2539}{7}a^{3}-\frac{426}{7}a^{2}+\frac{274}{7}a+\frac{29}{7}$, $\frac{18}{7}a^{21}-25a^{20}+\frac{517}{7}a^{19}+28a^{18}-\frac{3797}{7}a^{17}+\frac{4990}{7}a^{16}+\frac{7321}{7}a^{15}-\frac{21481}{7}a^{14}+\frac{3959}{7}a^{13}+\frac{34337}{7}a^{12}-\frac{32751}{7}a^{11}-\frac{18689}{7}a^{10}+\frac{47610}{7}a^{9}-\frac{9187}{7}a^{8}-\frac{31672}{7}a^{7}+\frac{15009}{7}a^{6}+\frac{10051}{7}a^{5}-\frac{5567}{7}a^{4}-178a^{3}+\frac{559}{7}a^{2}+\frac{31}{7}a+\frac{3}{7}$, $\frac{6}{7}a^{21}-\frac{66}{7}a^{20}+34a^{19}-\frac{86}{7}a^{18}-\frac{1459}{7}a^{17}+410a^{16}+243a^{15}-1447a^{14}+\frac{5564}{7}a^{13}+\frac{13778}{7}a^{12}-\frac{18915}{7}a^{11}-\frac{4023}{7}a^{10}+\frac{23708}{7}a^{9}-\frac{9290}{7}a^{8}-\frac{15172}{7}a^{7}+\frac{10665}{7}a^{6}+\frac{5057}{7}a^{5}-\frac{4295}{7}a^{4}-\frac{761}{7}a^{3}+\frac{625}{7}a^{2}+\frac{43}{7}a-3$, $\frac{3}{7}a^{21}-\frac{18}{7}a^{20}-\frac{26}{7}a^{19}+\frac{388}{7}a^{18}-89a^{17}-\frac{1468}{7}a^{16}+\frac{4806}{7}a^{15}-\frac{151}{7}a^{14}-\frac{12442}{7}a^{13}+\frac{10607}{7}a^{12}+\frac{12352}{7}a^{11}-\frac{23662}{7}a^{10}+215a^{9}+\frac{23776}{7}a^{8}-\frac{12648}{7}a^{7}-\frac{12584}{7}a^{6}+1351a^{5}+\frac{3484}{7}a^{4}-\frac{2553}{7}a^{3}-\frac{362}{7}a^{2}+\frac{195}{7}a-\frac{4}{7}$ Copy content Toggle raw display (assuming GRH)
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:  \( 6449194007.94 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{1}\cdot 6449194007.94 \cdot 1}{2\cdot\sqrt{15891297445653813609959315666834999}}\cr\approx \mathstrut & 0.168529351433 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*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 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*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 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*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 - 11*x^21 + 41*x^20 - 25*x^19 - 223*x^18 + 525*x^17 + 92*x^16 - 1654*x^15 + 1480*x^14 + 1772*x^13 - 3759*x^12 + 506*x^11 + 3901*x^10 - 2800*x^9 - 1867*x^8 + 2565*x^7 + 335*x^6 - 1066*x^5 - 4*x^4 + 202*x^3 + 3*x^2 - 14*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

$C_2^{10}.(C_2\times S_{11})$ (as 22T53):

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 81749606400
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed

Intermediate fields

11.11.104215884017569.1

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 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 ${\href{/padicField/2.11.0.1}{11} }^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ $18{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $22$ ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(79\) Copy content Toggle raw display 79.2.1.2$x^{2} + 79$$2$$1$$1$$C_2$$[\ ]_{2}$
79.20.0.1$x^{20} - x + 70$$1$$20$$0$20T1$[\ ]^{20}$
\(1451\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(18521\) Copy content Toggle raw display $\Q_{18521}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{18521}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(71823490019\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$