Normalized defining polynomial
\( x^{16} + 8x^{14} + 41x^{12} + 144x^{10} + 320x^{8} + 438x^{6} + 365x^{4} + 182x^{2} + 49 \)
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: | \(162447943996702457856\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{8}\) | 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: | \(18.33\) | 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: | $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{6}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{5502}a^{14}-\frac{419}{5502}a^{12}+\frac{176}{917}a^{10}-\frac{719}{2751}a^{8}-\frac{313}{917}a^{6}+\frac{443}{2751}a^{4}+\frac{1681}{5502}a^{2}+\frac{63}{262}$, $\frac{1}{38514}a^{15}-\frac{1585}{19257}a^{13}-\frac{1658}{6419}a^{11}-\frac{8972}{19257}a^{9}-\frac{2147}{6419}a^{7}+\frac{5945}{19257}a^{5}-\frac{3821}{38514}a^{3}-\frac{34}{917}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{83}{786} a^{14} - \frac{77}{131} a^{12} - \frac{1118}{393} a^{10} - \frac{3203}{393} a^{8} - \frac{5248}{393} a^{6} - \frac{5329}{393} a^{4} - \frac{2317}{262} a^{2} - \frac{998}{393} \) (order $6$) | 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: | $\frac{33}{1834}a^{14}+\frac{701}{5502}a^{12}+\frac{1837}{2751}a^{10}+\frac{1949}{917}a^{8}+\frac{12494}{2751}a^{6}+\frac{4532}{917}a^{4}+\frac{10529}{5502}a^{2}+\frac{371}{786}$, $\frac{1412}{19257}a^{15}+\frac{15263}{38514}a^{13}+\frac{36710}{19257}a^{11}+\frac{101569}{19257}a^{9}+\frac{156118}{19257}a^{7}+\frac{131375}{19257}a^{5}+\frac{60898}{19257}a^{3}+\frac{697}{5502}a$, $\frac{403}{6419}a^{15}+\frac{6161}{12838}a^{13}+\frac{15669}{6419}a^{11}+\frac{54133}{6419}a^{9}+\frac{117067}{6419}a^{7}+\frac{157152}{6419}a^{5}+\frac{122658}{6419}a^{3}+\frac{12557}{1834}a$, $\frac{1637}{19257}a^{15}+\frac{33}{1834}a^{14}+\frac{13781}{38514}a^{13}+\frac{701}{5502}a^{12}+\frac{32222}{19257}a^{11}+\frac{1837}{2751}a^{10}+\frac{69625}{19257}a^{9}+\frac{1949}{917}a^{8}+\frac{49933}{19257}a^{7}+\frac{12494}{2751}a^{6}-\frac{24154}{19257}a^{5}+\frac{4532}{917}a^{4}-\frac{67061}{19257}a^{3}+\frac{10529}{5502}a^{2}-\frac{14075}{5502}a+\frac{1157}{786}$, $\frac{1199}{38514}a^{15}+\frac{373}{786}a^{14}+\frac{1879}{12838}a^{13}+\frac{2485}{786}a^{12}+\frac{12263}{19257}a^{11}+\frac{1982}{131}a^{10}+\frac{26492}{19257}a^{9}+\frac{18703}{393}a^{8}+\frac{12136}{19257}a^{7}+\frac{11238}{131}a^{6}-\frac{54806}{19257}a^{5}+\frac{33977}{393}a^{4}-\frac{46477}{12838}a^{3}+\frac{37513}{786}a^{2}-\frac{3425}{5502}a+\frac{3625}{262}$, $\frac{1681}{38514}a^{15}+\frac{233}{2751}a^{14}+\frac{1973}{6419}a^{13}+\frac{1409}{2751}a^{12}+\frac{28339}{19257}a^{11}+\frac{2237}{917}a^{10}+\frac{92584}{19257}a^{9}+\frac{19825}{2751}a^{8}+\frac{174845}{19257}a^{7}+\frac{10949}{917}a^{6}+\frac{191732}{19257}a^{5}+\frac{27623}{2751}a^{4}+\frac{88499}{12838}a^{3}+\frac{12035}{2751}a^{2}+\frac{6436}{2751}a+\frac{269}{131}$, $\frac{3611}{12838}a^{15}+\frac{83}{393}a^{14}+\frac{23883}{12838}a^{13}+\frac{154}{131}a^{12}+\frac{57019}{6419}a^{11}+\frac{2236}{393}a^{10}+\frac{178533}{6419}a^{9}+\frac{6406}{393}a^{8}+\frac{318536}{6419}a^{7}+\frac{10496}{393}a^{6}+\frac{316790}{6419}a^{5}+\frac{10658}{393}a^{4}+\frac{337007}{12838}a^{3}+\frac{2317}{131}a^{2}+\frac{12545}{1834}a+\frac{2389}{393}$ (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: | \( 15253.0597214 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15253.0597214 \cdot 1}{6\cdot\sqrt{162447943996702457856}}\cr\approx \mathstrut & 0.484493056789 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times D_4$ (as 16T25):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
Character table for $C_2^2 \times D_4$ |
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
\(3\) | 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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7\) | 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |