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Properties

Label 16.8.34182052039...7929.1
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{14}$
Root discriminant $607.24$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13163633135399, 20205755303303, 6015138745506, -2927083507477, -1924503835524, -388988455119, -14134091903, 5120653975, 999608855, 100353656, -1657428, -728094, -65432, -4412, 144, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 144*x^14 - 4412*x^13 - 65432*x^12 - 728094*x^11 - 1657428*x^10 + 100353656*x^9 + 999608855*x^8 + 5120653975*x^7 - 14134091903*x^6 - 388988455119*x^5 - 1924503835524*x^4 - 2927083507477*x^3 + 6015138745506*x^2 + 20205755303303*x - 13163633135399)
 
gp: K = bnfinit(x^16 - 3*x^15 + 144*x^14 - 4412*x^13 - 65432*x^12 - 728094*x^11 - 1657428*x^10 + 100353656*x^9 + 999608855*x^8 + 5120653975*x^7 - 14134091903*x^6 - 388988455119*x^5 - 1924503835524*x^4 - 2927083507477*x^3 + 6015138745506*x^2 + 20205755303303*x - 13163633135399, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 144 x^{14} - 4412 x^{13} - 65432 x^{12} - 728094 x^{11} - 1657428 x^{10} + 100353656 x^{9} + 999608855 x^{8} + 5120653975 x^{7} - 14134091903 x^{6} - 388988455119 x^{5} - 1924503835524 x^{4} - 2927083507477 x^{3} + 6015138745506 x^{2} + 20205755303303 x - 13163633135399 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(341820520390299824133982419608164514148827929=37^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $607.24$
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:  $37, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{15} - \frac{1495279784688726930528397463516340410103579470114080483574211744187848044063904329064759743373448164811}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{14} + \frac{1566473349015877496542551540490398240959822778405728866284913697753344117801638888460509643333199630908}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{13} + \frac{3167118304801430406197678615762711690181907787051998808189018623208208604681790125105123809252871545738}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{12} + \frac{700448396279823990108457665328214809532694366166412430344412252918637458522434766531232237126810317699}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{11} - \frac{1295745600686477859847547982937245594164453798837469225359736818793229283743727567749613760186645834533}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{10} - \frac{3285376739541894689057404064500265531130013674087987361798835507603514054124999378922748317676590962826}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{9} - \frac{218302930049937905358486784326232000347160622016857570673327592421415512471057170699554379354804118398}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{8} + \frac{644385827957545870166555814511252845951344826078265096877803859225885163290592860793992972046970420937}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{7} - \frac{1707033858556410559881814345135431515902690610375343535665592862401876302637458112323207674842933972292}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{6} - \frac{3183325868020395505445453224194526005203960719994441090594024289647475743927356435567494500079238159382}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{5} + \frac{3006051843238457549750615160318581465876169273096821798894382116523003746973643171890671625770556217227}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{4} - \frac{3643428282596495333991836415501343330409594585172241962390831537731067932434426497146318449055617095959}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{3} - \frac{1345840080700376005989384907875305605450255067744613246621553074383603109330443028989100426932449883228}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a^{2} + \frac{168544011967580525222713021420981970468413714692864095935751173674230102913773477014869574578791279923}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629} a + \frac{1407345111594876800438619173401031553200747040603405291630630685750017963984604462986107332863900946831}{7463757183387817044541646703290722889560498744136509177767099322173249314768200059019451198792776182629}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 3177119625570000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{1517}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{41})\), 8.8.12187467896636600569.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$