Normalized defining polynomial
\( x^{16} - 3 x^{15} + 2 x^{14} + 6 x^{13} - 10 x^{12} - 3 x^{11} + 23 x^{10} - 24 x^{9} + 7 x^{8} + 6 x^{7} + x^{6} - 15 x^{5} + 20 x^{4} - 15 x^{3} + 7 x^{2} - 3 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(6158959248447369\)\(\medspace = 3^{12}\cdot 7^{4}\cdot 13^{6}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.70$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $3, 7, 13$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $8$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{238663} a^{15} + \frac{107485}{238663} a^{14} - \frac{89485}{238663} a^{13} + \frac{32552}{238663} a^{12} - \frac{88877}{238663} a^{11} - \frac{495}{14039} a^{10} + \frac{21273}{238663} a^{9} - \frac{38003}{238663} a^{8} + \frac{89451}{238663} a^{7} - \frac{107187}{238663} a^{6} - \frac{98593}{238663} a^{5} + \frac{27453}{238663} a^{4} + \frac{38752}{238663} a^{3} - \frac{10378}{238663} a^{2} + \frac{405}{238663} a + \frac{95971}{238663}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( \frac{8884}{1717} a^{15} - \frac{19678}{1717} a^{14} + \frac{1713}{1717} a^{13} + \frac{56036}{1717} a^{12} - \frac{44856}{1717} a^{11} - \frac{3878}{101} a^{10} + \frac{155389}{1717} a^{9} - \frac{85641}{1717} a^{8} - \frac{15313}{1717} a^{7} + \frac{45251}{1717} a^{6} + \frac{46225}{1717} a^{5} - \frac{98399}{1717} a^{4} + \frac{96684}{1717} a^{3} - \frac{50196}{1717} a^{2} + \frac{16358}{1717} a - \frac{9477}{1717} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 33.7844971841 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_4.C_2^2:D_4$ (as 16T211):
A solvable group of order 128 |
The 44 conjugacy class representatives for $C_4.C_2^2:D_4$ |
Character table for $C_4.C_2^2:D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.2, 8.0.8719893.2, 8.0.1601613.1, 8.0.6036849.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
3 | Data not computed | ||||||
$7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
$13$ | 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.3.4 | $x^{4} + 104$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.4 | $x^{4} + 104$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |