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Properties

Label 16.12.9693633931...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{56}\cdot 5^{6}\cdot 41^{6}\cdot 13463^{2}$
Root discriminant $273.31$
Ramified primes $2, 5, 41, 13463$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1127

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7617130807225, 0, -563817515520, 0, -123855275832, 0, 8558318760, 0, 284605746, 0, -13850960, 0, 157048, 0, -680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 680*x^14 + 157048*x^12 - 13850960*x^10 + 284605746*x^8 + 8558318760*x^6 - 123855275832*x^4 - 563817515520*x^2 + 7617130807225)
 
gp: K = bnfinit(x^16 - 680*x^14 + 157048*x^12 - 13850960*x^10 + 284605746*x^8 + 8558318760*x^6 - 123855275832*x^4 - 563817515520*x^2 + 7617130807225, 1)
 

Normalized defining polynomial

\( x^{16} - 680 x^{14} + 157048 x^{12} - 13850960 x^{10} + 284605746 x^{8} + 8558318760 x^{6} - 123855275832 x^{4} - 563817515520 x^{2} + 7617130807225 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(969363393189495136925157683101696000000=2^{56}\cdot 5^{6}\cdot 41^{6}\cdot 13463^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $273.31$
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:  $2, 5, 41, 13463$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{67093890937253689320301768428963995811617992780} a^{14} + \frac{1357243636614120150057807808443399502290617}{17028906329252205411244103662173602997872587} a^{12} + \frac{2488721580305985968182017209001636609600483303}{67093890937253689320301768428963995811617992780} a^{10} + \frac{1560567927241467306269504212527994180905416071}{13418778187450737864060353685792799162323598556} a^{8} + \frac{3968929345961059191410485606173976906531883463}{9584841562464812760043109775566285115945427540} a^{6} - \frac{416212415504645519543172597218566454318573317}{6709389093725368932030176842896399581161799278} a^{4} - \frac{6602588764546560797477856474384931960741446437}{67093890937253689320301768428963995811617992780} a^{2} + \frac{7025098924019908430366080770479874143695}{24310129455890376812438704970611049909732}$, $\frac{1}{67093890937253689320301768428963995811617992780} a^{15} + \frac{1357243636614120150057807808443399502290617}{17028906329252205411244103662173602997872587} a^{13} + \frac{2488721580305985968182017209001636609600483303}{67093890937253689320301768428963995811617992780} a^{11} + \frac{1560567927241467306269504212527994180905416071}{13418778187450737864060353685792799162323598556} a^{9} + \frac{3968929345961059191410485606173976906531883463}{9584841562464812760043109775566285115945427540} a^{7} - \frac{416212415504645519543172597218566454318573317}{6709389093725368932030176842896399581161799278} a^{5} - \frac{6602588764546560797477856474384931960741446437}{67093890937253689320301768428963995811617992780} a^{3} + \frac{7025098924019908430366080770479874143695}{24310129455890376812438704970611049909732} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 47290537393100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1127:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1127 are not computed
Character table for t16n1127 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.44066406400.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13463Data not computed