↑

Properties

Label 16.0.66753251506...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{9}$
Root discriminant $54.83$
Ramified primes $5, 101$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1281

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3798311, -6443580, 6879627, -5708255, 4545883, -2584630, 1511429, -654640, 289810, -94175, 30524, -6870, 1643, -245, 52, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 52*x^14 - 245*x^13 + 1643*x^12 - 6870*x^11 + 30524*x^10 - 94175*x^9 + 289810*x^8 - 654640*x^7 + 1511429*x^6 - 2584630*x^5 + 4545883*x^4 - 5708255*x^3 + 6879627*x^2 - 6443580*x + 3798311)
 
gp: K = bnfinit(x^16 - 5*x^15 + 52*x^14 - 245*x^13 + 1643*x^12 - 6870*x^11 + 30524*x^10 - 94175*x^9 + 289810*x^8 - 654640*x^7 + 1511429*x^6 - 2584630*x^5 + 4545883*x^4 - 5708255*x^3 + 6879627*x^2 - 6443580*x + 3798311, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 52 x^{14} - 245 x^{13} + 1643 x^{12} - 6870 x^{11} + 30524 x^{10} - 94175 x^{9} + 289810 x^{8} - 654640 x^{7} + 1511429 x^{6} - 2584630 x^{5} + 4545883 x^{4} - 5708255 x^{3} + 6879627 x^{2} - 6443580 x + 3798311 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6675325150661382452392578125=5^{14}\cdot 101^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{44} a^{14} + \frac{9}{44} a^{13} + \frac{13}{44} a^{12} - \frac{2}{11} a^{11} + \frac{1}{22} a^{10} - \frac{1}{2} a^{9} + \frac{5}{22} a^{8} + \frac{15}{44} a^{7} - \frac{3}{22} a^{6} - \frac{15}{44} a^{5} + \frac{17}{44} a^{4} + \frac{3}{44} a^{3} - \frac{3}{11} a^{2} - \frac{3}{22} a + \frac{1}{4}$, $\frac{1}{7857659068852756339934433020350436958824792} a^{15} + \frac{21335841556685785874167894729304958890893}{1964414767213189084983608255087609239706198} a^{14} - \frac{442375533444756371847301263845408231980011}{1964414767213189084983608255087609239706198} a^{13} - \frac{3102765895136493148904722303399566162043893}{7857659068852756339934433020350436958824792} a^{12} + \frac{1647538511471845878313735834698818435155185}{3928829534426378169967216510175218479412396} a^{11} - \frac{931586793253967626747019172435159644923595}{1964414767213189084983608255087609239706198} a^{10} + \frac{209917121390085028880172544943829817488223}{982207383606594542491804127543804619853099} a^{9} - \frac{515869623425148390462064729083763726320679}{7857659068852756339934433020350436958824792} a^{8} - \frac{2762245717036468939421771408756462337321901}{7857659068852756339934433020350436958824792} a^{7} + \frac{2617105691013084694892106746461995927010255}{7857659068852756339934433020350436958824792} a^{6} + \frac{650167150995324377582637720574266213891001}{1964414767213189084983608255087609239706198} a^{5} - \frac{326481617394331034895533201756223310526167}{3928829534426378169967216510175218479412396} a^{4} - \frac{644472088963480727579808999702275014323399}{7857659068852756339934433020350436958824792} a^{3} + \frac{1203265258608817014766299817614071762557555}{3928829534426378169967216510175218479412396} a^{2} - \frac{17671561090039294204350448063615712881711}{7857659068852756339934433020350436958824792} a - \frac{93220670229322738299848730561120146773721}{714332642622977849084948456395494268984072}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{458902357013625387647328219549983}{3314069619929462817348980607486476996552} a^{15} + \frac{1613054440649039590903397491244927}{1657034809964731408674490303743238498276} a^{14} - \frac{5839438776333704618470711109329315}{1657034809964731408674490303743238498276} a^{13} + \frac{157274828773734436433505531896352933}{3314069619929462817348980607486476996552} a^{12} - \frac{297980541821971959101270260768942893}{1657034809964731408674490303743238498276} a^{11} + \frac{1063188329924130572772580398596402173}{828517404982365704337245151871619249138} a^{10} - \frac{1599632085073743276208421533910472169}{414258702491182852168622575935809624569} a^{9} + \frac{70948286852474016285929933183700630969}{3314069619929462817348980607486476996552} a^{8} - \frac{219401234587595052468662155354408219547}{3314069619929462817348980607486476996552} a^{7} + \frac{892436864567510212080819435158187343339}{3314069619929462817348980607486476996552} a^{6} - \frac{1007634019602029282351995086120841100329}{1657034809964731408674490303743238498276} a^{5} + \frac{642795331215528773866457056505932265074}{414258702491182852168622575935809624569} a^{4} - \frac{7412133778505047638869727320446853343129}{3314069619929462817348980607486476996552} a^{3} + \frac{6702951035430644197487828849114546765581}{1657034809964731408674490303743238498276} a^{2} - \frac{13658548887851871818472487183687594553055}{3314069619929462817348980607486476996552} a + \frac{942319620953113829562910216016647064757}{301279056357223892486270964316952454232} \) (order $10$)
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:  \( 8261721.23087 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.159390625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. 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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$