Normalized defining polynomial
\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 166 x^{12} - 268 x^{11} + 428 x^{10} - 732 x^{9} + 1279 x^{8} - 1956 x^{7} + 2456 x^{6} - 2432 x^{5} + 1866 x^{4} - 1072 x^{3} + \cdots + 14 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(101415451701035401216\) \(\medspace = 2^{44}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(17.80\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2^{11/4}7^{1/2}\approx 17.798422345016238$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2381}a^{14}-\frac{7}{2381}a^{13}-\frac{1045}{2381}a^{12}-\frac{782}{2381}a^{11}+\frac{845}{2381}a^{10}-\frac{795}{2381}a^{9}+\frac{263}{2381}a^{8}+\frac{164}{2381}a^{7}+\frac{991}{2381}a^{6}-\frac{251}{2381}a^{5}-\frac{1001}{2381}a^{4}+\frac{846}{2381}a^{3}-\frac{49}{2381}a^{2}+\frac{820}{2381}a-\frac{1077}{2381}$, $\frac{1}{473819}a^{15}+\frac{92}{473819}a^{14}-\frac{77930}{473819}a^{13}-\frac{75665}{473819}a^{12}+\frac{125812}{473819}a^{11}+\frac{192386}{473819}a^{10}+\frac{76323}{473819}a^{9}+\frac{166680}{473819}a^{8}-\frac{94680}{473819}a^{7}-\frac{73574}{473819}a^{6}+\frac{83676}{473819}a^{5}-\frac{81586}{473819}a^{4}-\frac{35345}{473819}a^{3}-\frac{42127}{473819}a^{2}+\frac{58674}{473819}a-\frac{218530}{473819}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{70160}{473819}a^{15}-\frac{526200}{473819}a^{14}+\frac{1787000}{473819}a^{13}-\frac{3634800}{473819}a^{12}+\frac{5144700}{473819}a^{11}-\frac{5870590}{473819}a^{10}+\frac{9658994}{473819}a^{9}-\frac{21786948}{473819}a^{8}+\frac{37027286}{473819}a^{7}-\frac{41724227}{473819}a^{6}+\frac{19545587}{473819}a^{5}+\frac{13892638}{473819}a^{4}-\frac{38394197}{473819}a^{3}+\frac{34609151}{473819}a^{2}-\frac{17649996}{473819}a+\frac{3925721}{473819}$, $\frac{296745}{473819}a^{15}-\frac{2059124}{473819}a^{14}+\frac{7333570}{473819}a^{13}-\frac{17285964}{473819}a^{12}+\frac{31143492}{473819}a^{11}-\frac{46107604}{473819}a^{10}+\frac{75866596}{473819}a^{9}-\frac{131993881}{473819}a^{8}+\frac{233311373}{473819}a^{7}-\frac{326643549}{473819}a^{6}+\frac{366654588}{473819}a^{5}-\frac{297140089}{473819}a^{4}+\frac{174160806}{473819}a^{3}-\frac{60275899}{473819}a^{2}+\frac{6715532}{473819}a+\frac{2073567}{473819}$, $\frac{296745}{473819}a^{15}-\frac{2392051}{473819}a^{14}+\frac{9664059}{473819}a^{13}-\frac{25689137}{473819}a^{12}+\frac{51266173}{473819}a^{11}-\frac{82940315}{473819}a^{10}+\frac{131115563}{473819}a^{9}-\frac{222396596}{473819}a^{8}+\frac{389560800}{473819}a^{7}-\frac{603032659}{473819}a^{6}+\frac{759623072}{473819}a^{5}-\frac{747576788}{473819}a^{4}+\frac{552060612}{473819}a^{3}-\frac{294138908}{473819}a^{2}+\frac{98556022}{473819}a-\frac{16050159}{473819}$, $\frac{719612}{473819}a^{15}-\frac{5143564}{473819}a^{14}+\frac{18467195}{473819}a^{13}-\frac{43389628}{473819}a^{12}+\frac{77568447}{473819}a^{11}-\frac{114887200}{473819}a^{10}+\frac{188532185}{473819}a^{9}-\frac{333860417}{473819}a^{8}+\frac{584067625}{473819}a^{7}-\frac{818106735}{473819}a^{6}+\frac{908550521}{473819}a^{5}-\frac{744665450}{473819}a^{4}+\frac{447182817}{473819}a^{3}-\frac{172890089}{473819}a^{2}+\frac{35115269}{473819}a-\frac{964939}{473819}$, $\frac{106218}{473819}a^{15}-\frac{796635}{473819}a^{14}+\frac{2957177}{473819}a^{13}-\frac{7139353}{473819}a^{12}+\frac{13066417}{473819}a^{11}-\frac{19913465}{473819}a^{10}+\frac{32403853}{473819}a^{9}-\frac{57306273}{473819}a^{8}+\frac{99554603}{473819}a^{7}-\frac{143083489}{473819}a^{6}+\frac{165192257}{473819}a^{5}-\frac{146807069}{473819}a^{4}+\frac{99146661}{473819}a^{3}-\frac{47773077}{473819}a^{2}+\frac{16026361}{473819}a-\frac{2817093}{473819}$, $\frac{4600}{2381}a^{14}-\frac{32200}{2381}a^{13}+\frac{114527}{2381}a^{12}-\frac{268562}{2381}a^{11}+\frac{482170}{2381}a^{10}-\frac{716465}{2381}a^{9}+\frac{1183609}{2381}a^{8}-\frac{2074228}{2381}a^{7}+\frac{3634772}{2381}a^{6}-\frac{5080869}{2381}a^{5}+\frac{5702749}{2381}a^{4}-\frac{4703810}{2381}a^{3}+\frac{2879424}{2381}a^{2}-\frac{1125717}{2381}a+\frac{233999}{2381}$, $\frac{393376}{473819}a^{15}-\frac{2950320}{473819}a^{14}+\frac{10840640}{473819}a^{13}-\frac{25717640}{473819}a^{12}+\frac{45999170}{473819}a^{11}-\frac{68543739}{473819}a^{10}+\frac{111352831}{473819}a^{9}-\frac{200402877}{473819}a^{8}+\frac{348150024}{473819}a^{7}-\frac{492656783}{473819}a^{6}+\frac{546443319}{473819}a^{5}-\frac{459962818}{473819}a^{4}+\frac{287107412}{473819}a^{3}-\frac{125607886}{473819}a^{2}+\frac{34864665}{473819}a-\frac{4654687}{473819}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 4749.52173347 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4749.52173347 \cdot 2}{2\cdot\sqrt{101415451701035401216}}\cr\approx \mathstrut & 1.14560990923 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 16T39):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
Character table for $C_2^2\wr C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.22.99 | $x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
2.8.22.99 | $x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |