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Properties

Label 16.8.43753597243...401.17
Degree $16$
Signature $[8, 4]$
Discriminant $37^{10}\cdot 157^{10}$
Root discriminant $225.20$
Ramified primes $37, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2749255547, -7371959, -3831224257, 1377467087, 158786487, 461570699, -338054405, -30590271, 19130705, -967355, -137850, 15156, 1718, 323, -109, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 109*x^14 + 323*x^13 + 1718*x^12 + 15156*x^11 - 137850*x^10 - 967355*x^9 + 19130705*x^8 - 30590271*x^7 - 338054405*x^6 + 461570699*x^5 + 158786487*x^4 + 1377467087*x^3 - 3831224257*x^2 - 7371959*x + 2749255547)
 
gp: K = bnfinit(x^16 - 4*x^15 - 109*x^14 + 323*x^13 + 1718*x^12 + 15156*x^11 - 137850*x^10 - 967355*x^9 + 19130705*x^8 - 30590271*x^7 - 338054405*x^6 + 461570699*x^5 + 158786487*x^4 + 1377467087*x^3 - 3831224257*x^2 - 7371959*x + 2749255547, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 109 x^{14} + 323 x^{13} + 1718 x^{12} + 15156 x^{11} - 137850 x^{10} - 967355 x^{9} + 19130705 x^{8} - 30590271 x^{7} - 338054405 x^{6} + 461570699 x^{5} + 158786487 x^{4} + 1377467087 x^{3} - 3831224257 x^{2} - 7371959 x + 2749255547 \)

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:  \(43753597243831965200444502399484946401=37^{10}\cdot 157^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.20$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{217} a^{14} - \frac{103}{217} a^{13} + \frac{87}{217} a^{12} - \frac{40}{217} a^{11} - \frac{14}{31} a^{10} + \frac{12}{217} a^{9} - \frac{32}{217} a^{8} - \frac{67}{217} a^{7} + \frac{65}{217} a^{6} + \frac{48}{217} a^{5} + \frac{2}{217} a^{4} - \frac{44}{217} a^{3} - \frac{30}{217} a^{2} + \frac{29}{217} a + \frac{16}{217}$, $\frac{1}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{15} - \frac{5658910987290376361144664519038164149580793055296448248619912789693646645}{5172835393224495879511891752915391908835127527637517208194144908020408153159} a^{14} + \frac{13880259986165860886562167680361554930107718046107278049254101570604996505583}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{13} + \frac{3890272780857178909906498492845414202430589567846223811930715630282355459510}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{12} - \frac{10454382341066490447350856890971605913666639647711104491670624991836521414747}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{11} + \frac{9424535719793003634335057472459022973621554232799576925207348927252751381868}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{10} - \frac{570499874357846133679796132742539972390437776040537836352646921223260677081}{5172835393224495879511891752915391908835127527637517208194144908020408153159} a^{9} + \frac{15569456473159693460346616931700580152219994889916509421936081268797514267987}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{8} + \frac{14523910088621275849020061836640203640441838606862788883672073216854690429457}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{7} + \frac{334918510274746696727290104259333729554965340562813568353916304304354880359}{1168059604921660359889782008722830431027286861079439369592226269552995389423} a^{6} - \frac{17178478503452029363186695523766808411682209843323576474479506450662697368788}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{5} + \frac{8944998994417962309716187865570244973637207499669456346316306424166307351993}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{4} - \frac{14834883154458053899688608019404504048240619984191286536781794259218212759778}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{3} + \frac{13213303004826465789320335998750600254461737307803470458477958706676590361901}{36209847752571471156583242270407743361845892693462620457359014356142857072113} a^{2} - \frac{2581467610253398803101488971985741841022530282065658807526926062998428822758}{5172835393224495879511891752915391908835127527637517208194144908020408153159} a - \frac{16794227691311218767845096517723759545835044729181846421794374466282238424610}{36209847752571471156583242270407743361845892693462620457359014356142857072113}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 6277570089570 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{157}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{5809}) \), 4.4.912013.1 x2, 4.4.214933.1 x2, \(\Q(\sqrt{37}, \sqrt{157})\), 8.8.1138689997959361.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed
157Data not computed