Properties

Label 28.2.335...384.1
Degree $28$
Signature $[2, 13]$
Discriminant $-3.357\times 10^{51}$
Root discriminant $69.22$
Ramified primes $2, 29$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 319*x^24 - 928*x^22 - 2494*x^20 - 21982*x^18 + 156890*x^16 - 105676*x^14 - 41383*x^12 - 3321805*x^10 + 13163303*x^8 - 22596220*x^6 + 20965492*x^4 - 8196560*x^2 - 1114064)
 
gp: K = bnfinit(x^28 - 29*x^26 + 319*x^24 - 928*x^22 - 2494*x^20 - 21982*x^18 + 156890*x^16 - 105676*x^14 - 41383*x^12 - 3321805*x^10 + 13163303*x^8 - 22596220*x^6 + 20965492*x^4 - 8196560*x^2 - 1114064, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1114064, 0, -8196560, 0, 20965492, 0, -22596220, 0, 13163303, 0, -3321805, 0, -41383, 0, -105676, 0, 156890, 0, -21982, 0, -2494, 0, -928, 0, 319, 0, -29, 0, 1]);
 

\( x^{28} - 29 x^{26} + 319 x^{24} - 928 x^{22} - 2494 x^{20} - 21982 x^{18} + 156890 x^{16} - 105676 x^{14} - 41383 x^{12} - 3321805 x^{10} + 13163303 x^{8} - 22596220 x^{6} + 20965492 x^{4} - 8196560 x^{2} - 1114064 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-3356956934459190134013901917828042334392800598032384\)\(\medspace = -\,2^{40}\cdot 29^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.22$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{4} a^{6} + \frac{5}{12} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{4} a^{7} + \frac{5}{12} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{24} a^{14} - \frac{1}{12} a^{11} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} + \frac{1}{24} a^{8} - \frac{5}{12} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{168} a^{15} + \frac{1}{84} a^{13} - \frac{1}{56} a^{11} - \frac{11}{168} a^{9} - \frac{2}{21} a^{7} - \frac{5}{24} a^{5} - \frac{13}{28} a^{3} + \frac{5}{42} a$, $\frac{1}{504} a^{16} + \frac{1}{56} a^{14} - \frac{1}{168} a^{12} - \frac{1}{12} a^{11} - \frac{1}{28} a^{10} - \frac{1}{12} a^{9} + \frac{19}{504} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{37}{168} a^{4} + \frac{1}{12} a^{3} - \frac{9}{28} a^{2} - \frac{1}{6} a + \frac{7}{18}$, $\frac{1}{1008} a^{17} - \frac{1}{1008} a^{16} + \frac{1}{84} a^{14} - \frac{1}{48} a^{13} - \frac{13}{336} a^{12} + \frac{1}{112} a^{11} + \frac{13}{336} a^{10} - \frac{25}{504} a^{9} - \frac{41}{504} a^{8} + \frac{23}{112} a^{7} + \frac{17}{48} a^{6} + \frac{71}{168} a^{5} - \frac{43}{168} a^{4} - \frac{13}{28} a^{3} - \frac{25}{84} a^{2} + \frac{23}{126} a - \frac{4}{9}$, $\frac{1}{1008} a^{18} - \frac{1}{1008} a^{16} - \frac{1}{112} a^{14} - \frac{5}{168} a^{12} - \frac{11}{1008} a^{10} - \frac{43}{1008} a^{8} - \frac{1}{2} a^{7} - \frac{25}{112} a^{6} + \frac{47}{168} a^{4} + \frac{97}{252} a^{2} - \frac{1}{2} a + \frac{2}{9}$, $\frac{1}{7056} a^{19} + \frac{1}{7056} a^{17} + \frac{1}{2352} a^{15} + \frac{2}{49} a^{13} + \frac{139}{7056} a^{11} + \frac{565}{7056} a^{9} + \frac{1007}{2352} a^{7} - \frac{25}{56} a^{5} + \frac{85}{252} a^{3} + \frac{94}{441} a$, $\frac{1}{14112} a^{20} + \frac{1}{14112} a^{18} - \frac{11}{14112} a^{16} - \frac{11}{1176} a^{14} + \frac{181}{14112} a^{12} - \frac{1}{12} a^{11} + \frac{1111}{14112} a^{10} - \frac{1}{12} a^{9} - \frac{1067}{14112} a^{8} - \frac{13}{168} a^{6} + \frac{1}{4} a^{5} - \frac{1}{252} a^{4} + \frac{1}{12} a^{3} + \frac{545}{1764} a^{2} - \frac{1}{6} a - \frac{1}{9}$, $\frac{1}{28224} a^{21} - \frac{1}{28224} a^{20} + \frac{1}{28224} a^{19} - \frac{1}{28224} a^{18} - \frac{11}{28224} a^{17} + \frac{11}{28224} a^{16} - \frac{1}{588} a^{15} - \frac{19}{1176} a^{14} - \frac{827}{28224} a^{13} - \frac{181}{28224} a^{12} + \frac{2035}{28224} a^{11} - \frac{523}{28224} a^{10} - \frac{1991}{28224} a^{9} - \frac{1873}{28224} a^{8} + \frac{13}{336} a^{7} + \frac{83}{336} a^{6} - \frac{317}{1008} a^{5} - \frac{61}{1008} a^{4} + \frac{5}{882} a^{3} + \frac{683}{1764} a^{2} + \frac{1}{252} a - \frac{1}{36}$, $\frac{1}{592704} a^{22} - \frac{1}{296352} a^{20} + \frac{17}{42336} a^{18} - \frac{155}{592704} a^{16} - \frac{9335}{592704} a^{14} + \frac{3691}{148176} a^{12} - \frac{1}{12} a^{11} - \frac{3557}{148176} a^{10} - \frac{1}{12} a^{9} - \frac{9707}{592704} a^{8} - \frac{1}{2} a^{7} - \frac{7559}{21168} a^{6} + \frac{1}{4} a^{5} - \frac{14765}{148176} a^{4} + \frac{1}{12} a^{3} - \frac{3938}{9261} a^{2} + \frac{1}{3} a + \frac{37}{108}$, $\frac{1}{1185408} a^{23} - \frac{1}{1185408} a^{22} - \frac{1}{592704} a^{21} + \frac{1}{592704} a^{20} - \frac{1}{84672} a^{19} + \frac{25}{84672} a^{18} - \frac{407}{1185408} a^{17} + \frac{743}{1185408} a^{16} + \frac{493}{1185408} a^{15} + \frac{14627}{1185408} a^{14} + \frac{3187}{296352} a^{13} + \frac{3365}{296352} a^{12} - \frac{247}{9261} a^{11} + \frac{2249}{74088} a^{10} - \frac{70943}{1185408} a^{9} + \frac{6767}{1185408} a^{8} + \frac{311}{756} a^{7} - \frac{5245}{10584} a^{6} - \frac{69449}{296352} a^{5} - \frac{71671}{296352} a^{4} + \frac{1559}{9261} a^{3} - \frac{6383}{37044} a^{2} + \frac{3847}{10584} a + \frac{29}{216}$, $\frac{1}{2709842688} a^{24} + \frac{263}{387120384} a^{22} + \frac{9439}{338730336} a^{20} + \frac{1190699}{2709842688} a^{18} + \frac{16193}{677460672} a^{16} + \frac{3709877}{387120384} a^{14} - \frac{1495537}{169365168} a^{12} - \frac{119871835}{2709842688} a^{10} - \frac{34864273}{2709842688} a^{8} + \frac{132964487}{677460672} a^{6} - \frac{54108283}{677460672} a^{4} + \frac{32894249}{169365168} a^{2} + \frac{144913}{493776}$, $\frac{1}{5419685376} a^{25} - \frac{1}{5419685376} a^{24} + \frac{263}{774240768} a^{23} - \frac{263}{774240768} a^{22} + \frac{9439}{677460672} a^{21} - \frac{9439}{677460672} a^{20} - \frac{345493}{5419685376} a^{19} - \frac{1190699}{5419685376} a^{18} - \frac{367855}{1354921344} a^{17} - \frac{16193}{1354921344} a^{16} - \frac{1557067}{774240768} a^{15} - \frac{3709877}{774240768} a^{14} - \frac{4951969}{338730336} a^{13} + \frac{1495537}{338730336} a^{12} + \frac{215018021}{5419685376} a^{11} - \frac{331768613}{5419685376} a^{10} - \frac{96311953}{5419685376} a^{9} + \frac{34864273}{5419685376} a^{8} - \frac{559474057}{1354921344} a^{7} + \frac{544496185}{1354921344} a^{6} - \frac{29913259}{1354921344} a^{5} + \frac{392838619}{1354921344} a^{4} + \frac{102790985}{338730336} a^{3} + \frac{80015863}{338730336} a^{2} - \frac{11223839}{48390048} a - \frac{144913}{987552}$, $\frac{1}{2284351445439844188097536} a^{26} - \frac{5124095733929}{47590655113330087252032} a^{24} + \frac{1363382683578602263}{2284351445439844188097536} a^{22} - \frac{64842891516076821773}{2284351445439844188097536} a^{20} + \frac{850268112250719935161}{2284351445439844188097536} a^{18} - \frac{8779020968014583971}{207668313221804017099776} a^{16} + \frac{1913104556387043516923}{99319628062601921221632} a^{14} + \frac{6634133868292048118083}{326335920777120598299648} a^{12} - \frac{1}{12} a^{11} - \frac{75458226950711250595099}{1142175722719922094048768} a^{10} - \frac{1}{12} a^{9} - \frac{73264997973685096090595}{2284351445439844188097536} a^{8} - \frac{1}{2} a^{7} + \frac{98499428667056812542263}{285543930679980523512192} a^{6} + \frac{1}{4} a^{5} + \frac{216312965896651049804671}{571087861359961047024384} a^{4} + \frac{1}{12} a^{3} - \frac{1112504923048812373855}{23795327556665043626016} a^{2} + \frac{1}{3} a - \frac{565101812347605383}{59463542415656085696}$, $\frac{1}{2284351445439844188097536} a^{27} + \frac{29255821360699}{380725240906640698016256} a^{25} - \frac{1}{5419685376} a^{24} + \frac{212289333192903697}{2284351445439844188097536} a^{23} + \frac{2731}{5419685376} a^{22} - \frac{29161105111737130157}{2284351445439844188097536} a^{21} - \frac{5291}{338730336} a^{20} + \frac{84132664484690857039}{2284351445439844188097536} a^{19} + \frac{2585773}{5419685376} a^{18} - \frac{52721734691563140979}{207668313221804017099776} a^{17} + \frac{239363}{677460672} a^{16} - \frac{185962392169269475411}{99319628062601921221632} a^{15} - \frac{44453735}{5419685376} a^{14} - \frac{12745950036388619525117}{326335920777120598299648} a^{13} + \frac{6889679}{169365168} a^{12} + \frac{8253079932589187685145}{571087861359961047024384} a^{11} - \frac{297350597}{5419685376} a^{10} - \frac{84631962196405284443789}{2284351445439844188097536} a^{9} - \frac{22957811}{5419685376} a^{8} + \frac{28565790733108454842459}{142771965339990261756096} a^{7} - \frac{610784207}{1354921344} a^{6} + \frac{157372495364721533090641}{571087861359961047024384} a^{5} - \frac{69852353}{1354921344} a^{4} + \frac{112274626017172175101}{1982943963055420302168} a^{3} - \frac{122386577}{338730336} a^{2} - \frac{160356432824379480401}{2913713578367148199104} a + \frac{326003}{987552}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{7}$, which has order $7$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 764940809278458.8 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 764940809278458.8 \cdot 7}{2\sqrt{3356956934459190134013901917828042334392800598032384}}\approx 4.39664368151936$ (assuming GRH)

Galois group

$D_{28}$ (as 28T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 56
The 17 conjugacy class representatives for $D_{28}$
Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.2.390224.1, 7.1.38068692544.1, 14.2.42027535208278434566144.1

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

Sibling fields

Degree 28 sibling: Deg 28

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ $28$ $28$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
29Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.29.2t1.a.a$1$ $ 29 $ \(\Q(\sqrt{29}) \) $C_2$ (as 2T1) $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.116.2t1.a.a$1$ $ 2^{2} \cdot 29 $ \(\Q(\sqrt{-29}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.13456.4t3.c.a$2$ $ 2^{4} \cdot 29^{2}$ 4.2.390224.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.3364.7t2.a.c$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3364.14t3.a.c$2$ $ 2^{2} \cdot 29^{2}$ 14.2.42027535208278434566144.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3364.14t3.a.b$2$ $ 2^{2} \cdot 29^{2}$ 14.2.42027535208278434566144.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3364.7t2.a.b$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3364.7t2.a.a$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3364.14t3.a.a$2$ $ 2^{2} \cdot 29^{2}$ 14.2.42027535208278434566144.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.13456.28t10.a.b$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.13456.28t10.a.e$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.13456.28t10.a.a$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.13456.28t10.a.f$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.13456.28t10.a.c$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.13456.28t10.a.d$2$ $ 2^{4} \cdot 29^{2}$ 28.2.3356956934459190134013901917828042334392800598032384.1 $D_{28}$ (as 28T10) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.