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Properties

Label 20.0.288...625.1
Degree $20$
Signature $(0, 10)$
Discriminant $2.885\times 10^{23}$
Root discriminant \(14.89\)
Ramified primes $3,5,29$
Class number $1$
Class group trivial
Galois group $D_5^2$ (as 20T28)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + x + 1)
 
Copy content gp:K = bnfinit(y^20 - 3*y^19 + 3*y^18 - 6*y^17 + 12*y^16 - 4*y^15 - 6*y^14 + 7*y^13 - 34*y^12 + 26*y^11 + 68*y^10 - 96*y^9 + 19*y^8 + 44*y^7 - 12*y^6 + 26*y^5 + 23*y^4 + 4*y^3 + 5*y^2 + y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + x + 1)
 

\( x^{20} - 3 x^{19} + 3 x^{18} - 6 x^{17} + 12 x^{16} - 4 x^{15} - 6 x^{14} + 7 x^{13} - 34 x^{12} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(288467289442715712890625\) \(\medspace = 3^{10}\cdot 5^{10}\cdot 29^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.89\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{1/2}29^{4/5}\approx 57.27485888607036$
Ramified primes:   \(3\), \(5\), \(29\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $D_5$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-3}, \sqrt{5})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{15}a^{14}+\frac{1}{15}a^{13}-\frac{1}{15}a^{12}+\frac{1}{15}a^{11}-\frac{1}{15}a^{10}-\frac{1}{5}a^{9}+\frac{7}{15}a^{8}-\frac{7}{15}a^{7}+\frac{2}{15}a^{6}+\frac{2}{15}a^{5}-\frac{7}{15}a^{4}-\frac{1}{15}a^{3}+\frac{2}{15}a+\frac{1}{5}$, $\frac{1}{15}a^{15}-\frac{2}{15}a^{13}+\frac{2}{15}a^{12}-\frac{2}{15}a^{11}-\frac{2}{15}a^{10}-\frac{1}{3}a^{9}+\frac{1}{15}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{15}a^{3}+\frac{2}{15}a^{2}+\frac{1}{15}a-\frac{1}{5}$, $\frac{1}{45}a^{16}-\frac{1}{45}a^{15}+\frac{1}{45}a^{14}+\frac{7}{45}a^{13}-\frac{7}{45}a^{12}+\frac{1}{15}a^{11}+\frac{4}{45}a^{10}-\frac{2}{5}a^{9}-\frac{7}{15}a^{8}-\frac{4}{9}a^{7}-\frac{1}{15}a^{6}+\frac{7}{15}a^{5}+\frac{1}{5}a^{4}-\frac{17}{45}a^{3}-\frac{1}{45}a^{2}-\frac{2}{5}a-\frac{8}{45}$, $\frac{1}{45}a^{17}-\frac{1}{45}a^{14}+\frac{2}{15}a^{13}+\frac{1}{9}a^{12}-\frac{2}{45}a^{11}-\frac{1}{9}a^{10}+\frac{2}{5}a^{9}+\frac{1}{45}a^{8}+\frac{2}{9}a^{7}-\frac{2}{5}a^{5}-\frac{1}{9}a^{4}+\frac{2}{15}a^{3}+\frac{11}{45}a^{2}+\frac{16}{45}a-\frac{1}{9}$, $\frac{1}{2025}a^{18}+\frac{19}{2025}a^{17}-\frac{4}{675}a^{16}+\frac{8}{2025}a^{15}-\frac{16}{2025}a^{14}-\frac{7}{81}a^{13}+\frac{34}{675}a^{12}-\frac{274}{2025}a^{11}+\frac{127}{2025}a^{10}-\frac{578}{2025}a^{9}+\frac{536}{2025}a^{8}-\frac{94}{405}a^{7}-\frac{163}{675}a^{6}+\frac{556}{2025}a^{5}+\frac{127}{2025}a^{4}+\frac{347}{2025}a^{3}+\frac{197}{675}a^{2}+\frac{199}{405}a+\frac{487}{2025}$, $\frac{1}{3719925}a^{19}-\frac{64}{3719925}a^{18}+\frac{25951}{3719925}a^{17}-\frac{6871}{3719925}a^{16}-\frac{18676}{743985}a^{15}-\frac{79352}{3719925}a^{14}-\frac{389473}{3719925}a^{13}+\frac{22262}{148797}a^{12}+\frac{4604}{45925}a^{11}-\frac{599629}{3719925}a^{10}-\frac{5017}{16533}a^{9}+\frac{2119}{112725}a^{8}+\frac{771526}{3719925}a^{7}+\frac{772438}{3719925}a^{6}-\frac{859396}{3719925}a^{5}-\frac{594338}{1239975}a^{4}-\frac{221786}{743985}a^{3}-\frac{1832443}{3719925}a^{2}-\frac{121307}{413325}a+\frac{410434}{3719925}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -\frac{156544}{743985} a^{19} + \frac{628072}{743985} a^{18} - \frac{910432}{743985} a^{17} + \frac{1316017}{743985} a^{16} - \frac{2764099}{743985} a^{15} + \frac{2407223}{743985} a^{14} + \frac{619618}{743985} a^{13} - \frac{1968806}{743985} a^{12} + \frac{654008}{82665} a^{11} - \frac{9126962}{743985} a^{10} - \frac{281333}{27555} a^{9} + \frac{801974}{22545} a^{8} - \frac{2979599}{148797} a^{7} - \frac{7546483}{743985} a^{6} + \frac{7589557}{743985} a^{5} - \frac{847999}{247995} a^{4} - \frac{29816}{743985} a^{3} + \frac{3297802}{743985} a^{2} + \frac{11239}{82665} a + \frac{721814}{743985} \)  (order $6$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{429577}{3719925}a^{19}-\frac{663898}{1239975}a^{18}+\frac{3577298}{3719925}a^{17}-\frac{207358}{148797}a^{16}+\frac{3288254}{1239975}a^{15}-\frac{3815926}{1239975}a^{14}+\frac{2625874}{3719925}a^{13}+\frac{6283148}{3719925}a^{12}-\frac{19080898}{3719925}a^{11}+\frac{2301311}{247995}a^{10}+\frac{5091688}{3719925}a^{9}-\frac{7787462}{338175}a^{8}+\frac{86533982}{3719925}a^{7}-\frac{713992}{743985}a^{6}-\frac{1200374}{137775}a^{5}+\frac{628984}{148797}a^{4}-\frac{3208382}{3719925}a^{3}-\frac{8490742}{3719925}a^{2}+\frac{8351894}{3719925}a-\frac{1676764}{3719925}$, $\frac{481}{7515}a^{19}-\frac{5846}{22545}a^{18}+\frac{10402}{22545}a^{17}-\frac{5699}{7515}a^{16}+\frac{28841}{22545}a^{15}-\frac{30778}{22545}a^{14}+\frac{11426}{22545}a^{13}+\frac{7399}{7515}a^{12}-\frac{73759}{22545}a^{11}+\frac{96454}{22545}a^{10}+\frac{2033}{4509}a^{9}-\frac{220018}{22545}a^{8}+\frac{59453}{4509}a^{7}-\frac{7768}{2505}a^{6}-\frac{129086}{22545}a^{5}+\frac{126391}{22545}a^{4}+\frac{20201}{22545}a^{3}+\frac{4217}{7515}a^{2}+\frac{5851}{4509}a-\frac{211}{4509}$, $\frac{174386}{743985}a^{19}-\frac{1406276}{3719925}a^{18}-\frac{259513}{1239975}a^{17}-\frac{2574523}{3719925}a^{16}+\frac{4644317}{3719925}a^{15}+\frac{9010031}{3719925}a^{14}-\frac{384523}{247995}a^{13}-\frac{5170387}{3719925}a^{12}-\frac{21984416}{3719925}a^{11}-\frac{15534947}{3719925}a^{10}+\frac{82586858}{3719925}a^{9}+\frac{1146374}{338175}a^{8}-\frac{5892494}{247995}a^{7}+\frac{18887479}{3719925}a^{6}+\frac{65718019}{3719925}a^{5}+\frac{22932473}{3719925}a^{4}+\frac{13631771}{1239975}a^{3}+\frac{32417474}{3719925}a^{2}+\frac{1947368}{743985}a+\frac{166162}{413325}$, $\frac{358997}{3719925}a^{19}-\frac{891394}{3719925}a^{18}+\frac{108217}{413325}a^{17}-\frac{598474}{743985}a^{16}+\frac{4647037}{3719925}a^{15}-\frac{1968668}{3719925}a^{14}+\frac{98222}{137775}a^{13}-\frac{626357}{3719925}a^{12}-\frac{13872043}{3719925}a^{11}+\frac{1355887}{743985}a^{10}+\frac{13272133}{3719925}a^{9}-\frac{805022}{338175}a^{8}+\frac{727931}{137775}a^{7}-\frac{5597606}{743985}a^{6}+\frac{15903317}{3719925}a^{5}+\frac{5500169}{743985}a^{4}+\frac{1193486}{1239975}a^{3}+\frac{19089898}{3719925}a^{2}+\frac{8253584}{3719925}a+\frac{406172}{1239975}$, $\frac{211}{4509}a^{19}-\frac{574}{7515}a^{18}-\frac{2681}{22545}a^{17}+\frac{4072}{22545}a^{16}-\frac{493}{2505}a^{15}+\frac{8207}{7515}a^{14}-\frac{37108}{22545}a^{13}+\frac{18811}{22545}a^{12}-\frac{13673}{22545}a^{11}-\frac{15443}{7515}a^{10}+\frac{168194}{22545}a^{9}-\frac{18223}{4509}a^{8}-\frac{199973}{22545}a^{7}+\frac{68737}{4509}a^{6}-\frac{27524}{7515}a^{5}-\frac{101656}{22545}a^{4}+\frac{150656}{22545}a^{3}+\frac{24421}{22545}a^{2}+\frac{17926}{22545}a+\frac{6062}{4509}$, $\frac{163562}{3719925}a^{19}-\frac{1251739}{3719925}a^{18}+\frac{806291}{1239975}a^{17}-\frac{395353}{743985}a^{16}+\frac{4966387}{3719925}a^{15}-\frac{7578653}{3719925}a^{14}-\frac{983462}{1239975}a^{13}+\frac{8815228}{3719925}a^{12}-\frac{7264438}{3719925}a^{11}+\frac{5086723}{743985}a^{10}+\frac{3323008}{3719925}a^{9}-\frac{7556492}{338175}a^{8}+\frac{17309729}{1239975}a^{7}+\frac{9727918}{743985}a^{6}-\frac{49914928}{3719925}a^{5}-\frac{3664528}{743985}a^{4}+\frac{2804}{137775}a^{3}-\frac{16916627}{3719925}a^{2}-\frac{4087276}{3719925}a+\frac{135577}{1239975}$, $\frac{19397}{112725}a^{19}-\frac{139736}{338175}a^{18}+\frac{39853}{338175}a^{17}-\frac{16733}{37575}a^{16}+\frac{392429}{338175}a^{15}+\frac{14114}{13527}a^{14}-\frac{838678}{338175}a^{13}+\frac{107192}{112725}a^{12}-\frac{1454263}{338175}a^{11}-\frac{8093}{338175}a^{10}+\frac{5885191}{338175}a^{9}-\frac{822497}{67635}a^{8}-\frac{4320839}{338175}a^{7}+\frac{2245387}{112725}a^{6}+\frac{338962}{338175}a^{5}-\frac{1209713}{338175}a^{4}+\frac{3117146}{338175}a^{3}+\frac{13607}{7515}a^{2}-\frac{113848}{338175}a+\frac{90569}{67635}$, $\frac{216518}{743985}a^{19}-\frac{1952362}{3719925}a^{18}-\frac{388118}{3719925}a^{17}-\frac{3701521}{3719925}a^{16}+\frac{6980179}{3719925}a^{15}+\frac{8835382}{3719925}a^{14}-\frac{1370558}{743985}a^{13}-\frac{5135929}{3719925}a^{12}-\frac{9481844}{1239975}a^{11}-\frac{11415604}{3719925}a^{10}+\frac{32279297}{1239975}a^{9}+\frac{5179}{12525}a^{8}-\frac{18991108}{743985}a^{7}+\frac{22306393}{3719925}a^{6}+\frac{80208923}{3719925}a^{5}+\frac{2827289}{413325}a^{4}+\frac{36695621}{3719925}a^{3}+\frac{36258638}{3719925}a^{2}+\frac{757442}{247995}a+\frac{3957751}{3719925}$, $\frac{95336}{338175}a^{19}-\frac{48562}{67635}a^{18}+\frac{212207}{338175}a^{17}-\frac{611384}{338175}a^{16}+\frac{1046822}{338175}a^{15}-\frac{177121}{338175}a^{14}-\frac{84728}{338175}a^{13}+\frac{218548}{338175}a^{12}-\frac{374333}{37575}a^{11}+\frac{1413229}{338175}a^{10}+\frac{70938}{4175}a^{9}-\frac{1572818}{112725}a^{8}+\frac{1711376}{338175}a^{7}-\frac{533218}{338175}a^{6}+\frac{1230583}{338175}a^{5}+\frac{1661168}{112725}a^{4}+\frac{2731333}{338175}a^{3}+\frac{2461741}{338175}a^{2}+\frac{65251}{12525}a+\frac{320897}{338175}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 9642.05211515 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 9642.05211515 \cdot 1}{6\cdot\sqrt{288467289442715712890625}}\cr\approx \mathstrut & 0.286925152297 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + 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

$D_5^2$ (as 20T28):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.2.179030503125.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 sibling: data not computed
Minimal sibling: 10.2.179030503125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ R R ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{10}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ R ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(3\) Copy content Toggle raw display 3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.5.8a1.2$x^{10} + 120 x^{9} + 5770 x^{8} + 139200 x^{7} + 1693480 x^{6} + 8518464 x^{5} + 3386960 x^{4} + 556800 x^{3} + 46160 x^{2} + 1920 x + 61$$5$$2$$8$$D_5$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)