Normalized defining polynomial
\( x^{16} + 48x^{12} + 32x^{10} - 216x^{8} + 768x^{6} - 608x^{4} + 1152x^{2} - 648 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3965881151245791007623610368\) \(\medspace = -\,2^{79}\cdot 3^{8}\) | 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: | \(53.08\) | 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: | $2^{79/16}3^{1/2}\approx 53.07576114126626$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{7}-\frac{1}{3}a$, $\frac{1}{18}a^{8}-\frac{4}{9}a^{2}$, $\frac{1}{18}a^{9}-\frac{4}{9}a^{3}$, $\frac{1}{18}a^{10}-\frac{4}{9}a^{4}$, $\frac{1}{108}a^{11}-\frac{1}{54}a^{9}-\frac{1}{18}a^{7}+\frac{4}{27}a^{5}-\frac{11}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{540}a^{12}-\frac{1}{270}a^{10}+\frac{1}{90}a^{8}-\frac{19}{270}a^{6}+\frac{16}{135}a^{4}+\frac{4}{45}a^{2}-\frac{1}{5}$, $\frac{1}{540}a^{13}-\frac{1}{270}a^{11}+\frac{1}{90}a^{9}-\frac{19}{270}a^{7}+\frac{16}{135}a^{5}+\frac{4}{45}a^{3}-\frac{1}{5}a$, $\frac{1}{672300}a^{14}+\frac{149}{672300}a^{12}-\frac{2219}{168075}a^{10}+\frac{4829}{336150}a^{8}+\frac{9763}{336150}a^{6}+\frac{70798}{168075}a^{4}+\frac{265}{747}a^{2}-\frac{752}{2075}$, $\frac{1}{672300}a^{15}+\frac{149}{672300}a^{13}-\frac{2651}{672300}a^{11}-\frac{698}{168075}a^{9}-\frac{4456}{168075}a^{7}-\frac{16352}{168075}a^{5}-\frac{865}{2241}a^{3}+\frac{1894}{6225}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | 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: | $\frac{128}{18675}a^{14}+\frac{2}{2075}a^{12}-\frac{1984}{6225}a^{10}-\frac{3259}{18675}a^{8}+\frac{4128}{2075}a^{6}-\frac{33472}{6225}a^{4}+\frac{7184}{3735}a^{2}-\frac{19539}{2075}$, $\frac{103}{33615}a^{14}-\frac{158}{33615}a^{12}-\frac{1016}{6723}a^{10}-\frac{2195}{6723}a^{8}+\frac{2240}{6723}a^{6}-\frac{10691}{6723}a^{4}-\frac{88}{415}a^{2}-\frac{2173}{415}$, $\frac{272}{168075}a^{15}+\frac{799}{112050}a^{14}-\frac{6487}{672300}a^{13}+\frac{403}{24900}a^{12}-\frac{33181}{336150}a^{11}+\frac{39851}{112050}a^{10}-\frac{87176}{168075}a^{9}+\frac{54896}{56025}a^{8}-\frac{150547}{168075}a^{7}-\frac{9446}{18675}a^{6}+\frac{31051}{168075}a^{5}+\frac{53929}{56025}a^{4}+\frac{1267}{747}a^{3}-\frac{137}{83}a^{2}-\frac{197}{6225}a-\frac{813}{2075}$, $\frac{272}{168075}a^{15}-\frac{799}{112050}a^{14}-\frac{6487}{672300}a^{13}-\frac{403}{24900}a^{12}-\frac{33181}{336150}a^{11}-\frac{39851}{112050}a^{10}-\frac{87176}{168075}a^{9}-\frac{54896}{56025}a^{8}-\frac{150547}{168075}a^{7}+\frac{9446}{18675}a^{6}+\frac{31051}{168075}a^{5}-\frac{53929}{56025}a^{4}+\frac{1267}{747}a^{3}+\frac{137}{83}a^{2}-\frac{197}{6225}a+\frac{813}{2075}$, $\frac{31}{11205}a^{15}+\frac{1067}{168075}a^{14}-\frac{47}{7470}a^{13}+\frac{2981}{336150}a^{12}+\frac{1483}{11205}a^{11}+\frac{51068}{168075}a^{10}-\frac{1621}{7470}a^{9}+\frac{213337}{336150}a^{8}-\frac{354}{415}a^{7}-\frac{185093}{168075}a^{6}+\frac{37309}{11205}a^{5}+\frac{550919}{168075}a^{4}-\frac{89116}{11205}a^{3}+\frac{14503}{3735}a^{2}+\frac{2124}{415}a-\frac{5321}{2075}$, $\frac{287}{224100}a^{15}-\frac{313}{112050}a^{14}+\frac{433}{224100}a^{13}+\frac{43}{18675}a^{12}+\frac{7399}{112050}a^{11}-\frac{8626}{56025}a^{10}+\frac{13933}{112050}a^{9}+\frac{4933}{56025}a^{8}+\frac{988}{56025}a^{7}-\frac{2786}{6225}a^{6}+\frac{8096}{56025}a^{5}+\frac{33992}{56025}a^{4}+\frac{986}{3735}a^{3}-\frac{2168}{3735}a^{2}-\frac{1876}{6225}a+\frac{841}{2075}$, $\frac{287}{224100}a^{15}-\frac{313}{112050}a^{14}-\frac{433}{224100}a^{13}+\frac{43}{18675}a^{12}-\frac{7399}{112050}a^{11}-\frac{8626}{56025}a^{10}-\frac{13933}{112050}a^{9}+\frac{4933}{56025}a^{8}-\frac{988}{56025}a^{7}-\frac{2786}{6225}a^{6}-\frac{8096}{56025}a^{5}+\frac{33992}{56025}a^{4}-\frac{986}{3735}a^{3}-\frac{2168}{3735}a^{2}+\frac{1876}{6225}a+\frac{841}{2075}$, $\frac{587}{22410}a^{15}-\frac{2839}{168075}a^{14}-\frac{49}{7470}a^{13}-\frac{667}{336150}a^{12}-\frac{2825}{2241}a^{11}-\frac{277307}{336150}a^{10}-\frac{195}{166}a^{9}-\frac{221219}{336150}a^{8}+\frac{1310}{249}a^{7}+\frac{496066}{168075}a^{6}-\frac{44552}{2241}a^{5}-\frac{2366428}{168075}a^{4}+\frac{114334}{11205}a^{3}+\frac{42451}{3735}a^{2}-\frac{29174}{1245}a-\frac{43733}{2075}$ (assuming GRH) | 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: | \( 91169605.0717327 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 91169605.0717327 \cdot 1}{2\cdot\sqrt{3965881151245791007623610368}}\cr\approx \mathstrut & 1.11935875415302 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $D_{16}$ |
Character table for $D_{16}$ |
Intermediate fields
\(\Q(\sqrt{6}) \), 4.2.18432.2, 8.2.173946175488.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 16 sibling: | 16.0.1321960383748597002541203456.210 |
Minimal sibling: | 16.0.1321960383748597002541203456.210 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | $16$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16$ | $16$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.16.79.20 | $x^{16} + 32 x^{15} + 48 x^{12} + 32 x^{11} + 48 x^{10} + 8 x^{8} + 40 x^{4} + 48 x^{2} + 32 x + 42$ | $16$ | $1$ | $79$ | $D_{16}$ | $[3, 4, 5, 6]^{2}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |