Normalized defining polynomial
\( x^{16} + 94 x^{14} + 3466 x^{12} + 64966 x^{10} + 672448 x^{8} + 3914768 x^{6} + 12530639 x^{4} + \cdots + 12313081 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(157685109830355016729600000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 29^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(66.81\) | 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))
| |
Galois root discriminant: | $2^{7/4}5^{1/2}11^{1/2}29^{3/4}\approx 311.7326816854843$ | ||
Ramified primes: |
\(2\), \(5\), \(11\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{22}a^{10}-\frac{5}{22}a^{8}-\frac{5}{11}a^{6}-\frac{1}{2}a^{4}-\frac{2}{11}a^{2}-\frac{1}{2}$, $\frac{1}{22}a^{11}-\frac{5}{22}a^{9}-\frac{5}{11}a^{7}-\frac{1}{2}a^{5}-\frac{2}{11}a^{3}-\frac{1}{2}a$, $\frac{1}{7018}a^{12}+\frac{47}{3509}a^{10}-\frac{43}{7018}a^{8}-\frac{155}{638}a^{6}+\frac{2229}{7018}a^{4}+\frac{7}{22}a^{2}-\frac{1}{2}$, $\frac{1}{7018}a^{13}+\frac{47}{3509}a^{11}-\frac{43}{7018}a^{9}-\frac{155}{638}a^{7}+\frac{2229}{7018}a^{5}+\frac{7}{22}a^{3}-\frac{1}{2}a$, $\frac{1}{12\!\cdots\!58}a^{14}-\frac{93625747627}{12\!\cdots\!58}a^{12}-\frac{102474556121745}{62\!\cdots\!29}a^{10}-\frac{32505278458325}{11\!\cdots\!78}a^{8}+\frac{957884002583935}{62\!\cdots\!29}a^{6}-\frac{494580526682783}{11\!\cdots\!78}a^{4}-\frac{1707415994}{161201301071}a^{2}-\frac{19645195044}{161201301071}$, $\frac{1}{12\!\cdots\!58}a^{15}-\frac{93625747627}{12\!\cdots\!58}a^{13}-\frac{102474556121745}{62\!\cdots\!29}a^{11}-\frac{32505278458325}{11\!\cdots\!78}a^{9}+\frac{957884002583935}{62\!\cdots\!29}a^{7}-\frac{494580526682783}{11\!\cdots\!78}a^{5}-\frac{1707415994}{161201301071}a^{3}-\frac{19645195044}{161201301071}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{1654}$, which has order $13232$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: |
$\frac{374382}{14765599871}a^{14}+\frac{33591732}{14765599871}a^{12}+\frac{1155239530}{14765599871}a^{10}+\frac{19480821321}{14765599871}a^{8}+\frac{171266174872}{14765599871}a^{6}+\frac{769614010321}{14765599871}a^{4}+\frac{147247082110}{1342327261}a^{2}+\frac{9903238630}{122029751}$, $\frac{61161696654}{62\!\cdots\!29}a^{14}+\frac{5556357455051}{62\!\cdots\!29}a^{12}+\frac{193756755668006}{62\!\cdots\!29}a^{10}+\frac{300651582473243}{565655365458139}a^{8}+\frac{29\!\cdots\!16}{62\!\cdots\!29}a^{6}+\frac{11\!\cdots\!81}{565655365458139}a^{4}+\frac{73037133221024}{1773214311781}a^{2}+\frac{4283628103633}{161201301071}$, $\frac{11359136015}{62\!\cdots\!29}a^{14}+\frac{1138040651042}{62\!\cdots\!29}a^{12}+\frac{90053839963783}{12\!\cdots\!58}a^{10}+\frac{163673850337593}{11\!\cdots\!78}a^{8}+\frac{96\!\cdots\!64}{62\!\cdots\!29}a^{6}+\frac{331777711897845}{39010714859182}a^{4}+\frac{37691927379188}{1773214311781}a^{2}+\frac{5830617070947}{322402602142}$, $\frac{218925897072}{62\!\cdots\!29}a^{14}+\frac{19711879728119}{62\!\cdots\!29}a^{12}+\frac{680573538370476}{62\!\cdots\!29}a^{10}+\frac{10\!\cdots\!32}{565655365458139}a^{8}+\frac{10\!\cdots\!44}{62\!\cdots\!29}a^{6}+\frac{41\!\cdots\!70}{565655365458139}a^{4}+\frac{267550528688334}{1773214311781}a^{2}+\frac{17365806333863}{161201301071}$, $\frac{169123336433}{62\!\cdots\!29}a^{14}+\frac{15293562924110}{62\!\cdots\!29}a^{12}+\frac{10\!\cdots\!23}{12\!\cdots\!58}a^{10}+\frac{16\!\cdots\!71}{11\!\cdots\!78}a^{8}+\frac{81\!\cdots\!92}{62\!\cdots\!29}a^{6}+\frac{23\!\cdots\!27}{39010714859182}a^{4}+\frac{232205322846498}{1773214311781}a^{2}+\frac{32317376133549}{322402602142}$, $\frac{49802560639}{62\!\cdots\!29}a^{14}+\frac{4418316804009}{62\!\cdots\!29}a^{12}+\frac{297459671372229}{12\!\cdots\!58}a^{10}+\frac{39784483146263}{102846430083298}a^{8}+\frac{19\!\cdots\!52}{62\!\cdots\!29}a^{6}+\frac{13\!\cdots\!57}{11\!\cdots\!78}a^{4}+\frac{3213200531076}{161201301071}a^{2}+\frac{3059041738461}{322402602142}$, $\frac{64921904383}{12\!\cdots\!58}a^{14}+\frac{2889155914542}{62\!\cdots\!29}a^{12}+\frac{196534410904563}{12\!\cdots\!58}a^{10}+\frac{297060874471369}{11\!\cdots\!78}a^{8}+\frac{28\!\cdots\!21}{12\!\cdots\!58}a^{6}+\frac{11\!\cdots\!83}{11\!\cdots\!78}a^{4}+\frac{79808983456085}{3546428623562}a^{2}+\frac{3332558062483}{161201301071}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 3793.72993285 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3793.72993285 \cdot 13232}{2\cdot\sqrt{157685109830355016729600000000}}\cr\approx \mathstrut & 0.153534334952 \end{aligned}\] (assuming GRH)
Galois group
$C_4^2:D_4$ (as 16T305):
A solvable group of order 128 |
The 29 conjugacy class representatives for $C_4^2:D_4$ |
Character table for $C_4^2:D_4$ |
Intermediate fields
\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1 |
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 |
Minimal sibling: | 16.0.1357958936241444547272638464000000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ |
2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.8.6.1 | $x^{8} + 96 x^{7} + 3464 x^{6} + 55872 x^{5} + 345682 x^{4} + 114528 x^{3} + 113384 x^{2} + 1587648 x + 9488961$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |