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Properties

Label 16.0.58247709245...5641.2
Degree $16$
Signature $[0, 8]$
Discriminant $41^{9}\cdot 59^{12}$
Root discriminant $171.92$
Ramified primes $41, 59$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78854279291, -32602890292, -7312336230, 5658707593, -51797314, -479007815, 166298404, -21246331, -2696526, 1419676, -108941, -4679, 3106, -375, 75, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 75*x^14 - 375*x^13 + 3106*x^12 - 4679*x^11 - 108941*x^10 + 1419676*x^9 - 2696526*x^8 - 21246331*x^7 + 166298404*x^6 - 479007815*x^5 - 51797314*x^4 + 5658707593*x^3 - 7312336230*x^2 - 32602890292*x + 78854279291)
 
gp: K = bnfinit(x^16 - 2*x^15 + 75*x^14 - 375*x^13 + 3106*x^12 - 4679*x^11 - 108941*x^10 + 1419676*x^9 - 2696526*x^8 - 21246331*x^7 + 166298404*x^6 - 479007815*x^5 - 51797314*x^4 + 5658707593*x^3 - 7312336230*x^2 - 32602890292*x + 78854279291, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 75 x^{14} - 375 x^{13} + 3106 x^{12} - 4679 x^{11} - 108941 x^{10} + 1419676 x^{9} - 2696526 x^{8} - 21246331 x^{7} + 166298404 x^{6} - 479007815 x^{5} - 51797314 x^{4} + 5658707593 x^{3} - 7312336230 x^{2} - 32602890292 x + 78854279291 \)

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:  \(582477092451999490195840776529155641=41^{9}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $171.92$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{1127} a^{14} + \frac{15}{1127} a^{13} - \frac{318}{1127} a^{12} + \frac{277}{1127} a^{11} - \frac{251}{1127} a^{10} - \frac{233}{1127} a^{9} + \frac{158}{1127} a^{8} + \frac{54}{1127} a^{7} + \frac{349}{1127} a^{6} + \frac{116}{1127} a^{5} - \frac{496}{1127} a^{4} - \frac{334}{1127} a^{3} - \frac{32}{161} a^{2} - \frac{358}{1127} a - \frac{272}{1127}$, $\frac{1}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{15} + \frac{2593908627813756179843344040266253902001553207520231627567346175916856687}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{14} + \frac{5995762438494319972218564708516930019775878801075489428949144529237438764284}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{13} + \frac{3453535014792954415467929294886334989218419964803988478966242890513842462542}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{12} + \frac{2735626317187158182187696314125487483197356842437507614150328275662449799559}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{11} + \frac{3626880750013037020812623041385677172233154545709069164966855291703626405892}{15344784719363664829455425413967592029414487744242020864623580819621346728123} a^{10} - \frac{7349851041337909990586224619685670534778546959896417283607087920116193848552}{15344784719363664829455425413967592029414487744242020864623580819621346728123} a^{9} + \frac{12208817453124000374243577909394992327343230005002653562922854338441416388256}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{8} - \frac{3271404276480087406991638992630501469946452379476491857324394314217147388142}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{7} + \frac{39712633520941158072818122470116964974020158148092201797688390105434767093503}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{6} + \frac{34409243372082551894697979761056790936280956880978468208752810145069693128606}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{5} + \frac{50012043587133032206092329294021666699011736164903592363188848823925213213147}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{4} + \frac{46617933693364620611044217449297183350032225588529827087531526717047977303683}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{3} - \frac{41055231555912047524863316132602463859478858704601974304473361031135824829989}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a^{2} - \frac{43705199367801060405547686533609842375181053891255623137356668504377952622497}{107413493035545653806187977897773144205901414209694146052365065737349427096861} a + \frac{28667634535449352136667311728532824081916443462110676182691124413783041238462}{107413493035545653806187977897773144205901414209694146052365065737349427096861}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -1 \) (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:  \( 43814744973.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.142721.1, 8.0.835140637481.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ R

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
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$