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Properties

Label 16.8.389...625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.900\times 10^{19}$
Root discriminant \(16.77\)
Ramified primes $5,29,109$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:D_4$ (as 16T969)

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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^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 2*y^14 + 10*y^12 + 23*y^11 - 40*y^10 - 45*y^9 + 2*y^8 + 120*y^7 + 71*y^6 - 216*y^5 + 26*y^4 + 73*y^3 - 14*y^2 - 7*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*x + 1)
 

\( x^{16} - 2 x^{15} - 2 x^{14} + 10 x^{12} + 23 x^{11} - 40 x^{10} - 45 x^{9} + 2 x^{8} + 120 x^{7} + \cdots + 1 \) 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:  $16$
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:  $[8, 4]$
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:   \(38999408308687890625\) \(\medspace = 5^{8}\cdot 29^{4}\cdot 109^{4}\) 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:  \(16.77\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}29^{1/2}109^{1/2}\approx 125.7179382586272$
Ramified primes:   \(5\), \(29\), \(109\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{2335211601}a^{15}-\frac{79321355}{778403867}a^{14}+\frac{46470270}{778403867}a^{13}-\frac{280606577}{2335211601}a^{12}-\frac{905719504}{2335211601}a^{11}-\frac{69930790}{778403867}a^{10}+\frac{217132474}{778403867}a^{9}+\frac{274360907}{2335211601}a^{8}+\frac{491620276}{2335211601}a^{7}-\frac{1061763499}{2335211601}a^{6}+\frac{757967546}{2335211601}a^{5}-\frac{291370969}{778403867}a^{4}-\frac{493092518}{2335211601}a^{3}+\frac{850959046}{2335211601}a^{2}+\frac{341349535}{2335211601}a-\frac{292398184}{2335211601}$ 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:  $11$
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:   $\frac{178255}{270373}a^{15}-\frac{315861}{270373}a^{14}-\frac{377957}{270373}a^{13}-\frac{185483}{270373}a^{12}+\frac{1653211}{270373}a^{11}+\frac{4435251}{270373}a^{10}-\frac{5622204}{270373}a^{9}-\frac{8170132}{270373}a^{8}-\frac{3292203}{270373}a^{7}+\frac{18572339}{270373}a^{6}+\frac{16386231}{270373}a^{5}-\frac{29146941}{270373}a^{4}+\frac{1480229}{270373}a^{3}+\frac{4481091}{270373}a^{2}-\frac{327546}{270373}a+\frac{407357}{270373}$, $\frac{1745269223}{2335211601}a^{15}-\frac{1989359809}{2335211601}a^{14}-\frac{5085692423}{2335211601}a^{13}-\frac{4478743013}{2335211601}a^{12}+\frac{13190920225}{2335211601}a^{11}+\frac{51035309560}{2335211601}a^{10}-\frac{24701242123}{2335211601}a^{9}-\frac{96125451208}{2335211601}a^{8}-\frac{79791101980}{2335211601}a^{7}+\frac{43738826600}{778403867}a^{6}+\frac{76396167009}{778403867}a^{5}-\frac{167681610106}{2335211601}a^{4}-\frac{25695326403}{778403867}a^{3}+\frac{18372897659}{778403867}a^{2}-\frac{925379892}{778403867}a-\frac{6205383004}{2335211601}$, $\frac{3284857658}{2335211601}a^{15}-\frac{4717451266}{2335211601}a^{14}-\frac{8350107032}{2335211601}a^{13}-\frac{6080759684}{2335211601}a^{12}+\frac{27469703632}{2335211601}a^{11}+\frac{89342572447}{2335211601}a^{10}-\frac{73260218071}{2335211601}a^{9}-\frac{166690881292}{2335211601}a^{8}-\frac{108225859291}{2335211601}a^{7}+\frac{97208590581}{778403867}a^{6}+\frac{123572126058}{778403867}a^{5}-\frac{419423739523}{2335211601}a^{4}-\frac{21433747112}{778403867}a^{3}+\frac{31273958648}{778403867}a^{2}-\frac{1868384826}{778403867}a-\frac{2687040595}{2335211601}$, $\frac{500392879}{778403867}a^{15}-\frac{1595153977}{2335211601}a^{14}-\frac{4478743013}{2335211601}a^{13}-\frac{4261772005}{2335211601}a^{12}+\frac{3631372477}{778403867}a^{11}+\frac{45109526797}{2335211601}a^{10}-\frac{17588336173}{2335211601}a^{9}-\frac{83281640426}{2335211601}a^{8}-\frac{26071942320}{778403867}a^{7}+\frac{105274386194}{2335211601}a^{6}+\frac{209296542062}{2335211601}a^{5}-\frac{122462979007}{2335211601}a^{4}-\frac{72285960302}{2335211601}a^{3}+\frac{21657629446}{2335211601}a^{2}+\frac{6011501557}{2335211601}a-\frac{1745269223}{2335211601}$, $\frac{1013589024}{778403867}a^{15}-\frac{4323245434}{2335211601}a^{14}-\frac{7743157622}{2335211601}a^{13}-\frac{5863788676}{2335211601}a^{12}+\frac{8390966946}{778403867}a^{11}+\frac{83416789684}{2335211601}a^{10}-\frac{66147312121}{2335211601}a^{9}-\frac{153847070510}{2335211601}a^{8}-\frac{35550194757}{778403867}a^{7}+\frac{265683678137}{2335211601}a^{6}+\frac{350824419209}{2335211601}a^{5}-\frac{374205108424}{2335211601}a^{4}-\frac{59501222429}{2335211601}a^{3}+\frac{60360812413}{2335211601}a^{2}+\frac{3182486755}{2335211601}a+\frac{1773073186}{2335211601}$, $\frac{1820708782}{2335211601}a^{15}-\frac{1956293659}{2335211601}a^{14}-\frac{5211671732}{2335211601}a^{13}-\frac{1763049965}{778403867}a^{12}+\frac{12701900570}{2335211601}a^{11}+\frac{53919080923}{2335211601}a^{10}-\frac{20861990770}{2335211601}a^{9}-\frac{31643731096}{778403867}a^{8}-\frac{95128299611}{2335211601}a^{7}+\frac{119302685773}{2335211601}a^{6}+\frac{242076524527}{2335211601}a^{5}-\frac{141231025285}{2335211601}a^{4}-\frac{62037047695}{2335211601}a^{3}+\frac{12519283241}{2335211601}a^{2}+\frac{1580538014}{2335211601}a+\frac{1061151912}{778403867}$, $\frac{2162594669}{2335211601}a^{15}-\frac{2886721397}{2335211601}a^{14}-\frac{6405617923}{2335211601}a^{13}-\frac{1260006771}{778403867}a^{12}+\frac{18943262491}{2335211601}a^{11}+\frac{62519718653}{2335211601}a^{10}-\frac{46955960426}{2335211601}a^{9}-\frac{43249749393}{778403867}a^{8}-\frac{74247783895}{2335211601}a^{7}+\frac{210843928202}{2335211601}a^{6}+\frac{290664941327}{2335211601}a^{5}-\frac{294297985580}{2335211601}a^{4}-\frac{128312922008}{2335211601}a^{3}+\frac{101577123235}{2335211601}a^{2}+\frac{18294027133}{2335211601}a-\frac{3117363628}{778403867}$, $\frac{1569077999}{778403867}a^{15}-\frac{6933435230}{2335211601}a^{14}-\frac{11800932172}{2335211601}a^{13}-\frac{8150087681}{2335211601}a^{12}+\frac{13252203175}{778403867}a^{11}+\frac{127116897605}{2335211601}a^{10}-\frac{110693352440}{2335211601}a^{9}-\frac{236537384821}{2335211601}a^{8}-\frac{48797215011}{778403867}a^{7}+\frac{427569101578}{2335211601}a^{6}+\frac{521827313707}{2335211601}a^{5}-\frac{628064617439}{2335211601}a^{4}-\frac{75817881178}{2335211601}a^{3}+\frac{133504796315}{2335211601}a^{2}+\frac{2253376175}{2335211601}a-\frac{5229730036}{2335211601}$, $\frac{206407872}{778403867}a^{15}-\frac{184371254}{778403867}a^{14}-\frac{646652780}{778403867}a^{13}-\frac{769982307}{778403867}a^{12}+\frac{1429407285}{778403867}a^{11}+\frac{6625447844}{778403867}a^{10}-\frac{1019292488}{778403867}a^{9}-\frac{12017254021}{778403867}a^{8}-\frac{14817565072}{778403867}a^{7}+\frac{12186629546}{778403867}a^{6}+\frac{33980156256}{778403867}a^{5}-\frac{6997066555}{778403867}a^{4}-\frac{14457625034}{778403867}a^{3}-\frac{7992116155}{778403867}a^{2}+\frac{2770822919}{778403867}a+\frac{2353676043}{778403867}$, $\frac{1314957721}{778403867}a^{15}-\frac{3748208614}{2335211601}a^{14}-\frac{13395326936}{2335211601}a^{13}-\frac{11391124474}{2335211601}a^{12}+\frac{10287250833}{778403867}a^{11}+\frac{125097378946}{2335211601}a^{10}-\frac{40581400087}{2335211601}a^{9}-\frac{259214217041}{2335211601}a^{8}-\frac{72784587227}{778403867}a^{7}+\frac{315413125037}{2335211601}a^{6}+\frac{643281280298}{2335211601}a^{5}-\frac{332436703306}{2335211601}a^{4}-\frac{388505728433}{2335211601}a^{3}+\frac{123612352693}{2335211601}a^{2}+\frac{63450028366}{2335211601}a-\frac{10011213185}{2335211601}$, $\frac{1955025605}{2335211601}a^{15}-\frac{3208915525}{2335211601}a^{14}-\frac{4602434540}{2335211601}a^{13}-\frac{2421192821}{2335211601}a^{12}+\frac{17836978402}{2335211601}a^{11}+\frac{50392932460}{2335211601}a^{10}-\frac{55919889190}{2335211601}a^{9}-\frac{97270725133}{2335211601}a^{8}-\frac{42008882281}{2335211601}a^{7}+\frac{66648435833}{778403867}a^{6}+\frac{65866085483}{778403867}a^{5}-\frac{309136762546}{2335211601}a^{4}-\frac{7632585157}{778403867}a^{3}+\frac{26679674607}{778403867}a^{2}-\frac{713135881}{778403867}a-\frac{1580380324}{2335211601}$ 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:  \( 6055.40897697 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 6055.40897697 \cdot 1}{2\cdot\sqrt{38999408308687890625}}\cr\approx \mathstrut & 0.193438944766 \end{aligned}\] (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^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*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^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*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^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 2*x^14 + 10*x^12 + 23*x^11 - 40*x^10 - 45*x^9 + 2*x^8 + 120*x^7 + 71*x^6 - 216*x^5 + 26*x^4 + 73*x^3 - 14*x^2 - 7*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

$C_2^6:D_4$ (as 16T969):

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); G, transitive_group_identification(G)
 
A solvable group of order 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.57293125.2, 8.8.6244950625.1, 8.4.57293125.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) 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$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.1.2.1a1.1$x^{2} + 109$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.1.2.1a1.1$x^{2} + 109$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

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