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Properties

Label 21.1.116...000.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.168\times 10^{45}$
Root discriminant \(139.98\)
Ramified primes $2,3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_7^3.S_3$ (as 21T155)

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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^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864)
 
Copy content gp:K = bnfinit(y^21 - 28*y^15 - 24*y^14 + 196*y^9 + 336*y^8 + 144*y^7 + 1372*y^3 + 3528*y^2 + 3024*y + 864, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864)
 

\( x^{21} - 28x^{15} - 24x^{14} + 196x^{9} + 336x^{8} + 144x^{7} + 1372x^{3} + 3528x^{2} + 3024x + 864 \) 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:  $21$
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:  $[1, 10]$
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:   \(1167905317988463581983363308425635430400000000\) \(\medspace = 2^{24}\cdot 3^{18}\cdot 5^{8}\cdot 7^{28}\) 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:  \(139.98\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{8}$, $\frac{1}{4}a^{18}$, $\frac{1}{559872}a^{19}-\frac{4999}{46656}a^{18}+\frac{529}{5184}a^{17}+\frac{17}{648}a^{16}-\frac{43}{432}a^{15}-\frac{1}{12}a^{14}+\frac{11657}{139968}a^{13}+\frac{1}{23328}a^{12}+\frac{833}{3888}a^{11}-\frac{119}{648}a^{10}+\frac{17}{108}a^{9}-\frac{5}{18}a^{8}-\frac{46607}{139968}a^{7}+\frac{343}{139968}a+\frac{49}{23328}$, $\frac{1}{78364164096}a^{20}+\frac{6665}{13060694016}a^{19}+\frac{44422225}{2176782336}a^{18}+\frac{31731929}{362797056}a^{17}-\frac{2260319}{60466176}a^{16}+\frac{1129385}{10077696}a^{15}+\frac{1772006537}{19591041024}a^{14}+\frac{24949}{279936}a^{13}+\frac{3101}{46656}a^{12}-\frac{443}{7776}a^{11}-\frac{307}{1296}a^{10}+\frac{13}{216}a^{9}-\frac{3809369039}{19591041024}a^{8}-\frac{1088064569}{3265173504}a^{7}+\frac{343}{19591041024}a^{2}+\frac{1143121}{1632586752}a+\frac{326599}{544195584}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
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 $1$ (assuming GRH)
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:  $10$
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:   \( -1 \)  (order $2$) 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:   $\frac{77677399303}{78364164096}a^{20}-\frac{11488845025}{13060694016}a^{19}+\frac{1697274103}{2176782336}a^{18}-\frac{250469233}{362797056}a^{17}+\frac{36924391}{60466176}a^{16}-\frac{5438209}{10077696}a^{15}-\frac{534408087361}{19591041024}a^{14}+\frac{117649}{279936}a^{13}-\frac{16807}{46656}a^{12}+\frac{2401}{7776}a^{11}-\frac{343}{1296}a^{10}+\frac{49}{216}a^{9}+\frac{3802383196759}{19591041024}a^{8}+\frac{525074379601}{3265173504}a^{7}+\frac{26643347960929}{19591041024}a^{2}+\frac{3738951926983}{1632586752}a+\frac{525074379601}{544195584}$, $\frac{50724094814215}{78364164096}a^{20}-\frac{7421708903041}{13060694016}a^{19}+\frac{1085324861623}{2176782336}a^{18}-\frac{158649731665}{362797056}a^{17}+\frac{23198751847}{60466176}a^{16}-\frac{3394835617}{10077696}a^{15}-\frac{349266986098753}{19591041024}a^{14}+\frac{10772141}{93312}a^{13}-\frac{1471115}{15552}a^{12}+\frac{212381}{2592}a^{11}-\frac{34259}{432}a^{10}+\frac{5177}{72}a^{9}+\frac{24\cdots 95}{19591041024}a^{8}+\frac{346633584651697}{3265173504}a^{7}-\frac{51}{2}a^{6}-26a^{5}+\frac{109}{2}a^{4}+76a^{3}+\frac{17\cdots 01}{19591041024}a^{2}+\frac{24\cdots 03}{1632586752}a+\frac{346642835976625}{544195584}$, $\frac{5462797807}{26121388032}a^{20}+\frac{1053447719}{4353564672}a^{19}-\frac{941858273}{725594112}a^{18}+\frac{168010199}{120932352}a^{17}+\frac{42996559}{20155392}a^{16}-\frac{15353593}{3359232}a^{15}-\frac{16698707737}{6530347008}a^{14}-\frac{797849}{93312}a^{13}+\frac{93983}{15552}a^{12}+\frac{20455}{2592}a^{11}-\frac{22609}{432}a^{10}+\frac{1615}{72}a^{9}+\frac{845068607167}{6530347008}a^{8}+\frac{21979968649}{1088391168}a^{7}+\frac{15}{2}a^{6}+65a^{5}-250a^{4}-92a^{3}+\frac{5151973845817}{6530347008}a^{2}+\frac{484770913087}{544195584}a+\frac{49552544905}{181398528}$, $\frac{378482911}{15116544}a^{20}-\frac{4878157}{157464}a^{19}+\frac{15904273}{419904}a^{18}-\frac{1585223}{34992}a^{17}+\frac{612091}{11664}a^{16}-\frac{56315}{972}a^{15}-\frac{2426329705}{3779136}a^{14}+\frac{15035161}{69984}a^{13}-\frac{3360103}{11664}a^{12}+\frac{736345}{1944}a^{11}-\frac{159115}{324}a^{10}+\frac{16910}{27}a^{9}+\frac{15605074927}{3779136}a^{8}+\frac{1035589601}{314928}a^{7}-\frac{967}{2}a^{6}+617a^{5}-743a^{4}+861a^{3}+\frac{126059398153}{3779136}a^{2}+\frac{29481661469}{629856}a+\frac{913659977}{52488}$, $\frac{455337583431191}{39182082048}a^{20}-\frac{65035160979809}{6530347008}a^{19}+\frac{9282917988839}{1088391168}a^{18}-\frac{1324095075089}{181398528}a^{17}+\frac{188734614071}{30233088}a^{16}-\frac{26880874625}{5038848}a^{15}-\frac{31\cdots 21}{9795520512}a^{14}-\frac{5749063}{1458}a^{13}+\frac{1708373}{486}a^{12}-\frac{508021}{162}a^{11}+\frac{150707}{54}a^{10}-\frac{22436}{9}a^{9}+\frac{22\cdots 91}{9795520512}a^{8}+\frac{31\cdots 29}{1632586752}a^{7}+\frac{1317}{2}a^{6}-\frac{1195}{2}a^{5}+\frac{1017}{2}a^{4}-485a^{3}+\frac{15\cdots 37}{9795520512}a^{2}+\frac{22\cdots 63}{816293376}a+\frac{31\cdots 37}{272097792}$, $\frac{233307534999737}{6530347008}a^{20}-\frac{35443748217023}{1088391168}a^{19}+\frac{5375181173513}{181398528}a^{18}-\frac{813850448879}{30233088}a^{17}+\frac{123039414233}{5038848}a^{16}-\frac{18574464479}{839808}a^{15}-\frac{16\cdots 55}{1632586752}a^{14}+\frac{282356299}{7776}a^{13}-\frac{41224189}{1296}a^{12}+\frac{6018199}{216}a^{11}-\frac{878491}{36}a^{10}+\frac{128059}{6}a^{9}+\frac{11\cdots 41}{1632586752}a^{8}+\frac{15\cdots 03}{272097792}a^{7}-3780a^{6}+\frac{6603}{2}a^{5}-\frac{5767}{2}a^{4}+2379a^{3}+\frac{80\cdots 35}{1632586752}a^{2}+\frac{11\cdots 97}{136048896}a+\frac{15\cdots 43}{45349632}$, $\frac{14671349045}{19591041024}a^{20}-\frac{1557523427}{3265173504}a^{19}-\frac{459687355}{544195584}a^{18}+\frac{15507757}{90699264}a^{17}+\frac{16673621}{15116544}a^{16}+\frac{1693789}{2519424}a^{15}-\frac{112538015651}{4897760256}a^{14}-\frac{6307}{864}a^{13}+\frac{6229}{144}a^{12}+\frac{449}{24}a^{11}-\frac{239}{4}a^{10}-\frac{87}{2}a^{9}+\frac{1120512444197}{4897760256}a^{8}+\frac{227035443827}{816293376}a^{7}-\frac{745}{2}a^{6}-\frac{1315}{2}a^{5}+509a^{4}+1296a^{3}+\frac{1966274802179}{4897760256}a^{2}-\frac{128732935387}{408146688}a-\frac{20709595789}{136048896}$, $\frac{731338010495}{8707129344}a^{20}-\frac{200842253897}{1451188224}a^{19}+\frac{60979395503}{241864704}a^{18}-\frac{15529850009}{40310784}a^{17}+\frac{4180888031}{6718464}a^{16}-\frac{1061510249}{1119744}a^{15}-\frac{2079501310217}{2176782336}a^{14}-\frac{8316721}{31104}a^{13}-\frac{3962201}{5184}a^{12}+\frac{277391}{864}a^{11}-\frac{300545}{144}a^{10}+\frac{72023}{24}a^{9}+\frac{26252822475599}{2176782336}a^{8}+\frac{4037767365881}{362797056}a^{7}+\frac{9907}{2}a^{6}+12913a^{5}+2636a^{4}+13567a^{3}+\frac{203630175167273}{2176782336}a^{2}+\frac{19015394409743}{181398528}a+\frac{2041960294649}{60466176}$, $\frac{6907866715265}{39182082048}a^{20}-\frac{1967098066535}{6530347008}a^{19}-\frac{320581926415}{1088391168}a^{18}+\frac{71120193673}{181398528}a^{17}+\frac{20165244257}{30233088}a^{16}-\frac{2699125703}{5038848}a^{15}-\frac{60171117219575}{9795520512}a^{14}+\frac{342515411}{69984}a^{13}+\frac{208750315}{11664}a^{12}-\frac{8287369}{1944}a^{11}-\frac{10320497}{324}a^{10}-\frac{11252}{27}a^{9}+\frac{876007582601393}{9795520512}a^{8}+\frac{22182393264983}{1632586752}a^{7}-\frac{452589}{2}a^{6}-\frac{226941}{2}a^{5}+\frac{727549}{2}a^{4}+282702a^{3}-\frac{31\cdots 29}{9795520512}a^{2}-\frac{343862922570487}{816293376}a-\frac{33458884221097}{272097792}$, $\frac{61106511119837}{78364164096}a^{20}-\frac{13838557670651}{13060694016}a^{19}+\frac{3089027173229}{2176782336}a^{18}-\frac{679845429995}{362797056}a^{17}+\frac{147181144061}{60466176}a^{16}-\frac{31317538139}{10077696}a^{15}-\frac{351661552030907}{19591041024}a^{14}+\frac{580565591}{93312}a^{13}-\frac{135811409}{15552}a^{12}+\frac{31485719}{2592}a^{11}-\frac{7180649}{432}a^{10}+\frac{1625567}{72}a^{9}+\frac{24\cdots 53}{19591041024}a^{8}+\frac{307399815117323}{3265173504}a^{7}-\frac{34573}{2}a^{6}+\frac{45771}{2}a^{5}-30962a^{4}+39296a^{3}+\frac{19\cdots 07}{19591041024}a^{2}+\frac{22\cdots 93}{1632586752}a+\frac{265920139313675}{544195584}$ Copy content Toggle raw display (assuming GRH)
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:  \( 643181672097000 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 643181672097000 \cdot 1}{2\cdot\sqrt{1167905317988463581983363308425635430400000000}}\cr\approx \mathstrut & 1.80479768812845 \end{aligned}\] (assuming GRH)

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^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864) 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^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864, 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^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864); 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^21 - 28*x^15 - 24*x^14 + 196*x^9 + 336*x^8 + 144*x^7 + 1372*x^3 + 3528*x^2 + 3024*x + 864); 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

$A_7^3.S_3$ (as 21T155):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
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 non-solvable group of order 96018048000
The 228 conjugacy class representatives for $A_7^3.S_3$
Character table for $A_7^3.S_3$

Intermediate fields

3.1.140.1

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 42 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 R R R R $21$ $15{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ $15{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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
\(2\) Copy content Toggle raw display 2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.18.22a1.4$x^{18} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2$$18$$1$$22$18T434$$[\frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$$
\(3\) Copy content Toggle raw display 3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.6.3.18a116.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 27 x^{11} + 42 x^{10} + 42 x^{9} + 66 x^{8} + 57 x^{7} + 85 x^{6} + 78 x^{5} + 60 x^{4} + 50 x^{3} + 63 x^{2} + 42 x + 17$$3$$6$$18$18T530not computed
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(7\) Copy content Toggle raw display 7.1.7.7a1.3$x^{7} + 21 x + 7$$7$$1$$7$$F_7$$$[\frac{7}{6}]_{6}$$
7.1.14.21a2.16$x^{14} + 21 x^{9} + 21 x^{8} + 21$$14$$1$$21$14T14$$[\frac{7}{6}, \frac{5}{3}]_{6}$$

Spectrum of ring of integers

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