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Properties

Label 16.0.30341377703...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{14}\cdot 41^{4}$
Root discriminant $69.60$
Ramified primes $2, 5, 41$
Class number $15072$ (GRH)
Class group $[2, 2, 2, 2, 942]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32169161, -51389008, 62192636, -49012432, 34056956, -17888024, 8682804, -3186136, 1206042, -298656, 97564, -17024, 4636, -552, 116, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 116*x^14 - 552*x^13 + 4636*x^12 - 17024*x^11 + 97564*x^10 - 298656*x^9 + 1206042*x^8 - 3186136*x^7 + 8682804*x^6 - 17888024*x^5 + 34056956*x^4 - 49012432*x^3 + 62192636*x^2 - 51389008*x + 32169161)
 
gp: K = bnfinit(x^16 - 8*x^15 + 116*x^14 - 552*x^13 + 4636*x^12 - 17024*x^11 + 97564*x^10 - 298656*x^9 + 1206042*x^8 - 3186136*x^7 + 8682804*x^6 - 17888024*x^5 + 34056956*x^4 - 49012432*x^3 + 62192636*x^2 - 51389008*x + 32169161, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 116 x^{14} - 552 x^{13} + 4636 x^{12} - 17024 x^{11} + 97564 x^{10} - 298656 x^{9} + 1206042 x^{8} - 3186136 x^{7} + 8682804 x^{6} - 17888024 x^{5} + 34056956 x^{4} - 49012432 x^{3} + 62192636 x^{2} - 51389008 x + 32169161 \)

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:  \(303413777032806400000000000000=2^{44}\cdot 5^{14}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.60$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{4}$, $\frac{1}{44} a^{13} - \frac{5}{44} a^{12} - \frac{1}{44} a^{11} + \frac{1}{44} a^{10} + \frac{1}{44} a^{9} + \frac{1}{22} a^{8} - \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{19}{44} a^{5} - \frac{17}{44} a^{4} + \frac{1}{4} a^{3} + \frac{13}{44} a^{2} - \frac{7}{44} a + \frac{5}{22}$, $\frac{1}{44} a^{14} - \frac{1}{11} a^{12} - \frac{1}{11} a^{11} - \frac{5}{44} a^{10} - \frac{1}{11} a^{9} - \frac{1}{22} a^{8} + \frac{1}{11} a^{7} - \frac{7}{44} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{5}{11} a^{3} + \frac{3}{44} a^{2} + \frac{2}{11} a + \frac{3}{22}$, $\frac{1}{721458280780865384771024749688972332227499475117444} a^{15} - \frac{418999752427320625187095113408846652720717705617}{721458280780865384771024749688972332227499475117444} a^{14} + \frac{462442230928407455173259895311994120628205444805}{721458280780865384771024749688972332227499475117444} a^{13} + \frac{80383534723353998870083800208618221029771181829815}{721458280780865384771024749688972332227499475117444} a^{12} - \frac{4043925545780218883533957553159561565497904335702}{180364570195216346192756187422243083056874868779361} a^{11} - \frac{45013586769391887407622213478324398359291493198009}{721458280780865384771024749688972332227499475117444} a^{10} + \frac{3505763796220180904032404135437664750957150723089}{32793558217312062944137488622226015101249976141702} a^{9} - \frac{12970213324835816510430906960849260000218551212717}{180364570195216346192756187422243083056874868779361} a^{8} + \frac{16050847979564396557719675821587228287653241801017}{721458280780865384771024749688972332227499475117444} a^{7} - \frac{314326199631837219719939429540925186174943664659241}{721458280780865384771024749688972332227499475117444} a^{6} - \frac{330315275362657544026094450681209308052045716243687}{721458280780865384771024749688972332227499475117444} a^{5} - \frac{20639277200455670408347250517502864603036188083537}{721458280780865384771024749688972332227499475117444} a^{4} + \frac{33672987965795967011754243034148452911520176132043}{180364570195216346192756187422243083056874868779361} a^{3} + \frac{63863130010716476723040452504507495654456484294983}{721458280780865384771024749688972332227499475117444} a^{2} - \frac{140884934245523092684889760688269483993787620570517}{360729140390432692385512374844486166113749737558722} a - \frac{83100647417125303202146033988368013091065376876068}{180364570195216346192756187422243083056874868779361}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{942}$, which has order $15072$ (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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
2Data not computed
5Data not computed
41Data not computed