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Properties

Label 16.12.3972862448...6209.3
Degree $16$
Signature $[12, 2]$
Discriminant $71^{10}\cdot 73^{14}$
Root discriminant $612.98$
Ramified primes $71, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10463745523072, 5367379047776, -2159676870464, -1072055539736, 45775819228, 43159964896, 4096332547, -153576402, -159907014, -17262064, 1048841, 244538, 20409, -412, -284, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 284*x^14 - 412*x^13 + 20409*x^12 + 244538*x^11 + 1048841*x^10 - 17262064*x^9 - 159907014*x^8 - 153576402*x^7 + 4096332547*x^6 + 43159964896*x^5 + 45775819228*x^4 - 1072055539736*x^3 - 2159676870464*x^2 + 5367379047776*x + 10463745523072)
 
gp: K = bnfinit(x^16 - 4*x^15 - 284*x^14 - 412*x^13 + 20409*x^12 + 244538*x^11 + 1048841*x^10 - 17262064*x^9 - 159907014*x^8 - 153576402*x^7 + 4096332547*x^6 + 43159964896*x^5 + 45775819228*x^4 - 1072055539736*x^3 - 2159676870464*x^2 + 5367379047776*x + 10463745523072, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 284 x^{14} - 412 x^{13} + 20409 x^{12} + 244538 x^{11} + 1048841 x^{10} - 17262064 x^{9} - 159907014 x^{8} - 153576402 x^{7} + 4096332547 x^{6} + 43159964896 x^{5} + 45775819228 x^{4} - 1072055539736 x^{3} - 2159676870464 x^{2} + 5367379047776 x + 10463745523072 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(397286244885795491769071181517306466749106209=71^{10}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.98$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $71, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4544} a^{12} + \frac{7}{568} a^{11} + \frac{93}{4544} a^{10} - \frac{33}{1136} a^{9} + \frac{229}{2272} a^{8} + \frac{93}{2272} a^{7} - \frac{289}{4544} a^{6} - \frac{365}{2272} a^{5} - \frac{1}{64} a^{4} - \frac{383}{1136} a^{3} - \frac{13}{71} a^{2} - \frac{279}{568} a - \frac{31}{142}$, $\frac{1}{36352} a^{13} - \frac{1}{9088} a^{12} + \frac{567}{36352} a^{11} - \frac{797}{18176} a^{10} - \frac{3}{256} a^{9} + \frac{1831}{18176} a^{8} - \frac{1793}{36352} a^{7} + \frac{1063}{18176} a^{6} + \frac{8939}{36352} a^{5} - \frac{411}{18176} a^{4} + \frac{1943}{4544} a^{3} + \frac{1843}{4544} a^{2} + \frac{853}{2272} a - \frac{277}{568}$, $\frac{1}{34637428038656} a^{14} - \frac{138598303}{34637428038656} a^{13} + \frac{2358528651}{34637428038656} a^{12} - \frac{418549370743}{34637428038656} a^{11} - \frac{75507386245}{8659357009664} a^{10} - \frac{251916810977}{8659357009664} a^{9} + \frac{1382178228533}{34637428038656} a^{8} + \frac{1884957666873}{34637428038656} a^{7} - \frac{8024339465687}{34637428038656} a^{6} - \frac{522538868927}{34637428038656} a^{5} - \frac{2776691503791}{17318714019328} a^{4} + \frac{65627798305}{2164839252416} a^{3} + \frac{1835560490329}{4329678504832} a^{2} - \frac{237753947791}{2164839252416} a + \frac{165629981275}{541209813104}$, $\frac{1}{406558739862879005500172853196456659056426903093132462621483008} a^{15} + \frac{378761571954727787865257504471102697235510611733}{406558739862879005500172853196456659056426903093132462621483008} a^{14} - \frac{4077926849139687060829968762693999677844634432437178378375}{406558739862879005500172853196456659056426903093132462621483008} a^{13} + \frac{226026778894566219176192497075829309678972994538667113765}{2967582042794737266424619366397493861725743818198047172419584} a^{12} + \frac{1980586989021256553220435741496144145785390786438835634779307}{203279369931439502750086426598228329528213451546566231310741504} a^{11} - \frac{911019423837843925589970478547698188251203564069758816132037}{50819842482859875687521606649557082382053362886641557827685376} a^{10} - \frac{6666832272911004519515935509685619373823989051477968310132767}{406558739862879005500172853196456659056426903093132462621483008} a^{9} + \frac{38304562495168071810407800127759710307206013904632293406538569}{406558739862879005500172853196456659056426903093132462621483008} a^{8} + \frac{7010669771303536394046223274291047394960944652545556391303523}{406558739862879005500172853196456659056426903093132462621483008} a^{7} + \frac{36961749395094432756684210543396314146913836677576731646721809}{406558739862879005500172853196456659056426903093132462621483008} a^{6} - \frac{17838705732713111876579735915626929909934494836333690127182955}{101639684965719751375043213299114164764106725773283115655370752} a^{5} + \frac{441573051498014424812280002674297138520830208328057463073307}{2479016706480969545732761299978394262539188433494710137935872} a^{4} - \frac{11503096245988230860539139587331127396321136596260668625553153}{50819842482859875687521606649557082382053362886641557827685376} a^{3} + \frac{4241805195186786126359864662357690871461372988521456296758105}{12704960620714968921880401662389270595513340721660389456921344} a^{2} + \frac{3191234231782435299696267400590543699335177589176546282691221}{12704960620714968921880401662389270595513340721660389456921344} a - \frac{1120409786481169215634546484164004048167868885237451409240343}{3176240155178742230470100415597317648878335180415097364230336}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 20044029886600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.280732967047165372057.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.2$x^{4} - 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$