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Properties

Label 16.8.66296687817...9481.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{14}\cdot 17^{14}$
Root discriminant $112.55$
Ramified primes $13, 17$
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![-11550751, 105429551, -3454475, -116506574, 22835637, -293238, 4370455, 1865554, -762393, -73306, 279, 2043, 1237, -230, 24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 24*x^14 - 230*x^13 + 1237*x^12 + 2043*x^11 + 279*x^10 - 73306*x^9 - 762393*x^8 + 1865554*x^7 + 4370455*x^6 - 293238*x^5 + 22835637*x^4 - 116506574*x^3 - 3454475*x^2 + 105429551*x - 11550751)
 
gp: K = bnfinit(x^16 - 3*x^15 + 24*x^14 - 230*x^13 + 1237*x^12 + 2043*x^11 + 279*x^10 - 73306*x^9 - 762393*x^8 + 1865554*x^7 + 4370455*x^6 - 293238*x^5 + 22835637*x^4 - 116506574*x^3 - 3454475*x^2 + 105429551*x - 11550751, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 24 x^{14} - 230 x^{13} + 1237 x^{12} + 2043 x^{11} + 279 x^{10} - 73306 x^{9} - 762393 x^{8} + 1865554 x^{7} + 4370455 x^{6} - 293238 x^{5} + 22835637 x^{4} - 116506574 x^{3} - 3454475 x^{2} + 105429551 x - 11550751 \)

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:  \(662966878170355779548558581239481=13^{14}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $13, 17$
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}{146617447300162531986160830052766220366441163984991147179944514903} a^{15} + \frac{44483798010069848919670499981295559574116971691652394635595993024}{146617447300162531986160830052766220366441163984991147179944514903} a^{14} - \frac{16022725882399897025412442043771596209636524088299890477943188161}{146617447300162531986160830052766220366441163984991147179944514903} a^{13} + \frac{38041771118851265684447625215301768554795203044949853115513251661}{146617447300162531986160830052766220366441163984991147179944514903} a^{12} - \frac{63700430954239306469111635556469600938173849651868334713945524268}{146617447300162531986160830052766220366441163984991147179944514903} a^{11} - \frac{8559001332518025541637364522841718974557789484699902969824217677}{146617447300162531986160830052766220366441163984991147179944514903} a^{10} - \frac{69636434337425852268521235823629962534035150752140122290895242508}{146617447300162531986160830052766220366441163984991147179944514903} a^{9} - \frac{204404434821310559982262184427276602627986391321258817158473319}{146617447300162531986160830052766220366441163984991147179944514903} a^{8} - \frac{48218944000486580212690682666635226335546018238697569051490682428}{146617447300162531986160830052766220366441163984991147179944514903} a^{7} + \frac{31048420061844600113470286255883148081406823014551375980930369948}{146617447300162531986160830052766220366441163984991147179944514903} a^{6} + \frac{49793254725588333123525261785208776987851177780789528605407909655}{146617447300162531986160830052766220366441163984991147179944514903} a^{5} - \frac{61159554211852454390656595810536222112642034357120708056094972837}{146617447300162531986160830052766220366441163984991147179944514903} a^{4} - \frac{53075942757142671576456714051369090574007191484378874965511074049}{146617447300162531986160830052766220366441163984991147179944514903} a^{3} - \frac{7822834797312794917942845181126991050853619362933530143110468592}{146617447300162531986160830052766220366441163984991147179944514903} a^{2} + \frac{7266835662680166896612530146127282231230795960127619500687835073}{146617447300162531986160830052766220366441163984991147179944514903} a + \frac{30578067369331123047339172644266186843059005628153474355042992829}{146617447300162531986160830052766220366441163984991147179944514903}$

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:  \( 4437980972.81 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 8.8.116507435287321.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$