↑

Properties

Label 16.0.35687701699...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{9}$
Root discriminant $45.66$
Ramified primes $5, 61$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1192

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![309425941, -124179562, 45222615, 34785530, -23390105, 1958559, 2189112, -1401095, 61525, 175355, -26098, -10254, 2035, 295, -70, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 70*x^14 + 295*x^13 + 2035*x^12 - 10254*x^11 - 26098*x^10 + 175355*x^9 + 61525*x^8 - 1401095*x^7 + 2189112*x^6 + 1958559*x^5 - 23390105*x^4 + 34785530*x^3 + 45222615*x^2 - 124179562*x + 309425941)
 
gp: K = bnfinit(x^16 - 3*x^15 - 70*x^14 + 295*x^13 + 2035*x^12 - 10254*x^11 - 26098*x^10 + 175355*x^9 + 61525*x^8 - 1401095*x^7 + 2189112*x^6 + 1958559*x^5 - 23390105*x^4 + 34785530*x^3 + 45222615*x^2 - 124179562*x + 309425941, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 70 x^{14} + 295 x^{13} + 2035 x^{12} - 10254 x^{11} - 26098 x^{10} + 175355 x^{9} + 61525 x^{8} - 1401095 x^{7} + 2189112 x^{6} + 1958559 x^{5} - 23390105 x^{4} + 34785530 x^{3} + 45222615 x^{2} - 124179562 x + 309425941 \)

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:  \(356877016993229400634765625=5^{15}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.66$
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:  $5, 61$
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}{514552587477350281165161774326407486667599771207238865869} a^{15} - \frac{128729456600645302200352631774329223227898480664356264660}{514552587477350281165161774326407486667599771207238865869} a^{14} + \frac{133041353582439889544180057695094236850446408877715182784}{514552587477350281165161774326407486667599771207238865869} a^{13} - \frac{7015378904312154221660484533812319933322311365784438659}{16598470563785492940811670139561531827987089393781898899} a^{12} + \frac{209207090547462299145993398165844305435152288504320166361}{514552587477350281165161774326407486667599771207238865869} a^{11} - \frac{121302641513881642702409173198416874660058925487637953473}{514552587477350281165161774326407486667599771207238865869} a^{10} - \frac{180629150243875572017868991446575770382821780658331553714}{514552587477350281165161774326407486667599771207238865869} a^{9} - \frac{194320101335317734291913440050945994761768079517108441859}{514552587477350281165161774326407486667599771207238865869} a^{8} - \frac{191853455142806795046972258570934282299135613783714867427}{514552587477350281165161774326407486667599771207238865869} a^{7} + \frac{60263699211552639175636402676194945406795140645545069570}{514552587477350281165161774326407486667599771207238865869} a^{6} - \frac{116691815397149426262247051635525718537214317037856145983}{514552587477350281165161774326407486667599771207238865869} a^{5} + \frac{114420913630032173756246181936892794361204419311352784529}{514552587477350281165161774326407486667599771207238865869} a^{4} + \frac{125652376481431000757522680623891907016914003776289125739}{514552587477350281165161774326407486667599771207238865869} a^{3} + \frac{92557178113924344849280711846751159981548992152449769943}{514552587477350281165161774326407486667599771207238865869} a^{2} + \frac{165224546891392386278365960025602125409884920903921182907}{514552587477350281165161774326407486667599771207238865869} a + \frac{192692699095854861880270001702740195104169577784705124838}{514552587477350281165161774326407486667599771207238865869}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( \frac{5570593416481222535772709822452666324841614}{77108582436137957831709739058916997635368642688559} a^{15} - \frac{35767223282545253031524389286921761290556110}{77108582436137957831709739058916997635368642688559} a^{14} - \frac{272194046849594386500326869889629708230945101}{77108582436137957831709739058916997635368642688559} a^{13} + \frac{2561870639460510334995556425489019639221048384}{77108582436137957831709739058916997635368642688559} a^{12} + \frac{2840309309466882324872350266152626706723728802}{77108582436137957831709739058916997635368642688559} a^{11} - \frac{65860450726461054254646218738526845290941640731}{77108582436137957831709739058916997635368642688559} a^{10} + \frac{69492693973193911003399882262209167134671849502}{77108582436137957831709739058916997635368642688559} a^{9} + \frac{699770980093046561728259893195545541864349175626}{77108582436137957831709739058916997635368642688559} a^{8} - \frac{1807622011254908174764355139214451724508727413968}{77108582436137957831709739058916997635368642688559} a^{7} - \frac{1003214282031823107845551973842324146955045108783}{77108582436137957831709739058916997635368642688559} a^{6} + \frac{12201037683161773271587762242904192365132554340964}{77108582436137957831709739058916997635368642688559} a^{5} - \frac{34260027361796845897745131622936229816525327327990}{77108582436137957831709739058916997635368642688559} a^{4} + \frac{7232917310785238557046362801524106423475200041029}{77108582436137957831709739058916997635368642688559} a^{3} + \frac{141558276786549678460091703422232210458664561580552}{77108582436137957831709739058916997635368642688559} a^{2} - \frac{253423563135700102339111214851828652722764189787367}{77108582436137957831709739058916997635368642688559} a + \frac{491361336996994928280438881402739880233169080018336}{77108582436137957831709739058916997635368642688559} \) (order $10$)
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:  \( 2862996.09182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
61Data not computed