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Properties

Label 16.0.14390210149...481.10
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 89^{12}$
Root discriminant $242.59$
Ramified primes $17, 89$
Class number $724808$ (GRH)
Class group $[2, 602, 602]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6855061765612, -539518460270, 1014049127255, -70436280004, 68938613619, -4255788538, 2881145428, -159005322, 82648724, -3988680, 1674140, -67494, 23472, -728, 213, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 213*x^14 - 728*x^13 + 23472*x^12 - 67494*x^11 + 1674140*x^10 - 3988680*x^9 + 82648724*x^8 - 159005322*x^7 + 2881145428*x^6 - 4255788538*x^5 + 68938613619*x^4 - 70436280004*x^3 + 1014049127255*x^2 - 539518460270*x + 6855061765612)
 
gp: K = bnfinit(x^16 - 4*x^15 + 213*x^14 - 728*x^13 + 23472*x^12 - 67494*x^11 + 1674140*x^10 - 3988680*x^9 + 82648724*x^8 - 159005322*x^7 + 2881145428*x^6 - 4255788538*x^5 + 68938613619*x^4 - 70436280004*x^3 + 1014049127255*x^2 - 539518460270*x + 6855061765612, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 213 x^{14} - 728 x^{13} + 23472 x^{12} - 67494 x^{11} + 1674140 x^{10} - 3988680 x^{9} + 82648724 x^{8} - 159005322 x^{7} + 2881145428 x^{6} - 4255788538 x^{5} + 68938613619 x^{4} - 70436280004 x^{3} + 1014049127255 x^{2} - 539518460270 x + 6855061765612 \)

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:  \(143902101499474565788603458614754288481=17^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.59$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{5}{16} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{1446362960} a^{12} - \frac{3}{1446362960} a^{11} - \frac{26428027}{723181480} a^{10} - \frac{143396961}{1446362960} a^{9} + \frac{11214589}{1446362960} a^{8} - \frac{170411917}{723181480} a^{7} - \frac{39617113}{289272592} a^{6} + \frac{97724923}{1446362960} a^{5} + \frac{92623791}{723181480} a^{4} + \frac{404917697}{1446362960} a^{3} + \frac{575439699}{1446362960} a^{2} + \frac{6281547}{15386840} a + \frac{108202033}{361590740}$, $\frac{1}{1446362960} a^{13} + \frac{18770811}{723181480} a^{11} + \frac{59625617}{1446362960} a^{10} - \frac{28692777}{723181480} a^{9} + \frac{72404179}{723181480} a^{8} - \frac{316580217}{1446362960} a^{7} + \frac{56662427}{361590740} a^{6} + \frac{13216963}{723181480} a^{5} + \frac{237478963}{1446362960} a^{4} - \frac{37935165}{144636296} a^{3} - \frac{66633909}{144636296} a^{2} + \frac{53863829}{361590740} a - \frac{63691163}{180795370}$, $\frac{1}{5785451840} a^{14} - \frac{1}{2892725920} a^{12} + \frac{20463201}{1446362960} a^{11} - \frac{89536669}{2892725920} a^{10} + \frac{158440021}{2892725920} a^{9} + \frac{316635951}{2892725920} a^{8} - \frac{576117203}{2892725920} a^{7} - \frac{77304791}{413246560} a^{6} - \frac{157825621}{1446362960} a^{5} + \frac{293284421}{2892725920} a^{4} + \frac{886226051}{2892725920} a^{3} - \frac{518174771}{1157090368} a^{2} + \frac{907384467}{2892725920} a + \frac{412604673}{1446362960}$, $\frac{1}{950352717220912782631073553325267283545175680} a^{15} - \frac{19327891869890910839949029912657371}{950352717220912782631073553325267283545175680} a^{14} - \frac{30172940407500819177739329746504513}{95035271722091278263107355332526728354517568} a^{13} + \frac{134830668637762990597818982542223079}{475176358610456391315536776662633641772587840} a^{12} - \frac{3680297740141714662764224342972424039432997}{475176358610456391315536776662633641772587840} a^{11} - \frac{2107585004313604460375540743720149513865889}{59397044826307048914442097082829205221573480} a^{10} + \frac{4152528251563936675806091222805547357759085}{47517635861045639131553677666263364177258784} a^{9} + \frac{7305527531698014543771697830700069030805749}{237588179305228195657768388331316820886293920} a^{8} + \frac{129428510910422768413549333375049593553317}{848529211804386413063458529754702931736764} a^{7} - \frac{75883954660323010330188828202011828470568657}{475176358610456391315536776662633641772587840} a^{6} - \frac{32622697939141188474002369113025652746861883}{475176358610456391315536776662633641772587840} a^{5} - \frac{2766742676896829154177986301631057670241417}{11879408965261409782888419416565841044314696} a^{4} + \frac{19782672664349617132790115710767234400074347}{950352717220912782631073553325267283545175680} a^{3} + \frac{436400536913819732744149836398783842445854247}{950352717220912782631073553325267283545175680} a^{2} - \frac{17926323597491261641770382731359265149626701}{67882336944350913045076682380376234538941120} a + \frac{21015118624329242268167578944832099708258281}{237588179305228195657768388331316820886293920}$

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

Class group and class number

$C_{2}\times C_{602}\times C_{602}$, which has order $724808$ (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:  \( 22977967.6331 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.0.11984473.1 x2, 4.0.203736041.1 x2, 4.4.38915873.1, 4.4.4913.1, 8.0.41508374402353681.3, 8.8.1514445171352129.1, 8.0.11995920202280213809.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.3.2$x^{4} - 801$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.2$x^{4} - 801$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.2$x^{4} - 801$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.2$x^{4} - 801$$4$$1$$3$$C_4$$[\ ]_{4}$