↑

Properties

Label 16.0.36574060273...5664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 41^{4}\cdot 569^{3}\cdot 1279^{2}$
Root discriminant $81.32$
Ramified primes $2, 41, 569, 1279$
Class number $24696$ (GRH)
Class group $[2, 2, 6174]$ (GRH)
Galois group 16T1781

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9287425, -1797280, 9659452, -1825304, 6371904, -1357128, 2239790, -334160, 409672, -35572, 42462, -1720, 2501, -32, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 78*x^14 - 32*x^13 + 2501*x^12 - 1720*x^11 + 42462*x^10 - 35572*x^9 + 409672*x^8 - 334160*x^7 + 2239790*x^6 - 1357128*x^5 + 6371904*x^4 - 1825304*x^3 + 9659452*x^2 - 1797280*x + 9287425)
 
gp: K = bnfinit(x^16 + 78*x^14 - 32*x^13 + 2501*x^12 - 1720*x^11 + 42462*x^10 - 35572*x^9 + 409672*x^8 - 334160*x^7 + 2239790*x^6 - 1357128*x^5 + 6371904*x^4 - 1825304*x^3 + 9659452*x^2 - 1797280*x + 9287425, 1)
 

Normalized defining polynomial

\( x^{16} + 78 x^{14} - 32 x^{13} + 2501 x^{12} - 1720 x^{11} + 42462 x^{10} - 35572 x^{9} + 409672 x^{8} - 334160 x^{7} + 2239790 x^{6} - 1357128 x^{5} + 6371904 x^{4} - 1825304 x^{3} + 9659452 x^{2} - 1797280 x + 9287425 \)

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:  \(3657406027372345478303568625664=2^{32}\cdot 41^{4}\cdot 569^{3}\cdot 1279^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.32$
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:  $2, 41, 569, 1279$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{606072605} a^{14} + \frac{266246417}{606072605} a^{13} - \frac{260601523}{606072605} a^{12} + \frac{188640517}{606072605} a^{11} + \frac{11140775}{121214521} a^{10} + \frac{3277499}{121214521} a^{9} - \frac{214595798}{606072605} a^{8} - \frac{140882203}{606072605} a^{7} - \frac{290814334}{606072605} a^{6} + \frac{276066752}{606072605} a^{5} + \frac{265676284}{606072605} a^{4} - \frac{44115716}{121214521} a^{3} + \frac{63777889}{606072605} a^{2} + \frac{225497644}{606072605} a - \frac{55265565}{121214521}$, $\frac{1}{842870261665440265388515925403015764825} a^{15} + \frac{18844523562602888124074159115}{33714810466617610615540637016120630593} a^{14} + \frac{265428932974045828513414524942212453938}{842870261665440265388515925403015764825} a^{13} - \frac{298265028421853355125172904991535653437}{842870261665440265388515925403015764825} a^{12} + \frac{284302649733237041378861284289988999996}{842870261665440265388515925403015764825} a^{11} + \frac{15228437798799380102770952651563178331}{33714810466617610615540637016120630593} a^{10} + \frac{100901609371413937821309327908740449987}{842870261665440265388515925403015764825} a^{9} - \frac{273953249986569602320323716826824603872}{842870261665440265388515925403015764825} a^{8} + \frac{416718400896713759123490102712708559967}{842870261665440265388515925403015764825} a^{7} + \frac{22311565258118483610998599272439594702}{168574052333088053077703185080603152965} a^{6} + \frac{369931275933563858045105880599013976}{911211093692367854474071270705962989} a^{5} + \frac{238449268230938659951473146105402520092}{842870261665440265388515925403015764825} a^{4} + \frac{335958017897484762034944420210626746469}{842870261665440265388515925403015764825} a^{3} + \frac{301654605337949595200510913296842675071}{842870261665440265388515925403015764825} a^{2} - \frac{186979372320303074236523406161866203383}{842870261665440265388515925403015764825} a + \frac{24407554804810650696680860193356766187}{168574052333088053077703185080603152965}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6174}$, which has order $24696$ (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:  \( 14950.5276193 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1781 are not computed
Character table for t16n1781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.3917778944.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
569Data not computed
1279Data not computed