Normalized defining polynomial
\( x^{16} - 2 x^{15} - 17 x^{14} + 36 x^{13} + 141 x^{12} - 438 x^{11} + 255 x^{10} + 2168 x^{9} + \cdots + 52645 \)
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: | \(220346584964513535947265625\) \(\medspace = 5^{10}\cdot 41^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.30\) | 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: | $5^{3/4}41^{3/4}\approx 54.17705527224281$ | ||
Ramified primes: | \(5\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{80}a^{14}+\frac{9}{80}a^{13}+\frac{3}{80}a^{12}-\frac{1}{40}a^{11}+\frac{1}{40}a^{10}-\frac{9}{40}a^{9}+\frac{19}{80}a^{8}-\frac{1}{80}a^{7}+\frac{1}{10}a^{6}-\frac{1}{80}a^{5}+\frac{7}{20}a^{4}+\frac{3}{80}a^{3}+\frac{7}{40}a^{2}+\frac{5}{16}a+\frac{7}{16}$, $\frac{1}{90\!\cdots\!00}a^{15}+\frac{77\!\cdots\!89}{22\!\cdots\!00}a^{14}-\frac{44\!\cdots\!47}{45\!\cdots\!00}a^{13}+\frac{10\!\cdots\!59}{90\!\cdots\!00}a^{12}+\frac{36\!\cdots\!47}{22\!\cdots\!00}a^{11}+\frac{50\!\cdots\!29}{22\!\cdots\!00}a^{10}+\frac{12\!\cdots\!33}{90\!\cdots\!00}a^{9}+\frac{68\!\cdots\!79}{11\!\cdots\!50}a^{8}+\frac{14\!\cdots\!41}{90\!\cdots\!00}a^{7}-\frac{92\!\cdots\!01}{18\!\cdots\!20}a^{6}+\frac{31\!\cdots\!61}{90\!\cdots\!00}a^{5}+\frac{78\!\cdots\!19}{90\!\cdots\!00}a^{4}-\frac{76\!\cdots\!93}{18\!\cdots\!20}a^{3}-\frac{48\!\cdots\!23}{15\!\cdots\!00}a^{2}+\frac{25\!\cdots\!97}{90\!\cdots\!60}a-\frac{44\!\cdots\!43}{18\!\cdots\!20}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $5$ |
Class group and class number
$C_{2}$, which has order $2$ (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: | $\frac{56\!\cdots\!51}{46\!\cdots\!00}a^{15}-\frac{15\!\cdots\!99}{46\!\cdots\!00}a^{14}-\frac{84\!\cdots\!89}{46\!\cdots\!00}a^{13}+\frac{13\!\cdots\!97}{23\!\cdots\!00}a^{12}+\frac{28\!\cdots\!99}{23\!\cdots\!00}a^{11}-\frac{14\!\cdots\!47}{23\!\cdots\!00}a^{10}+\frac{39\!\cdots\!73}{46\!\cdots\!00}a^{9}+\frac{96\!\cdots\!87}{46\!\cdots\!00}a^{8}-\frac{23\!\cdots\!77}{23\!\cdots\!00}a^{7}-\frac{15\!\cdots\!09}{92\!\cdots\!60}a^{6}+\frac{12\!\cdots\!33}{23\!\cdots\!00}a^{5}-\frac{16\!\cdots\!21}{46\!\cdots\!00}a^{4}-\frac{26\!\cdots\!59}{23\!\cdots\!90}a^{3}+\frac{63\!\cdots\!47}{78\!\cdots\!00}a^{2}+\frac{20\!\cdots\!19}{92\!\cdots\!60}a-\frac{64\!\cdots\!86}{11\!\cdots\!45}$, $\frac{16\!\cdots\!51}{45\!\cdots\!00}a^{15}-\frac{73\!\cdots\!87}{22\!\cdots\!00}a^{14}-\frac{71\!\cdots\!91}{11\!\cdots\!50}a^{13}+\frac{29\!\cdots\!69}{45\!\cdots\!00}a^{12}+\frac{31\!\cdots\!56}{56\!\cdots\!25}a^{11}-\frac{58\!\cdots\!93}{56\!\cdots\!25}a^{10}-\frac{18\!\cdots\!77}{45\!\cdots\!00}a^{9}+\frac{18\!\cdots\!81}{22\!\cdots\!00}a^{8}-\frac{71\!\cdots\!79}{45\!\cdots\!00}a^{7}-\frac{36\!\cdots\!29}{90\!\cdots\!60}a^{6}+\frac{58\!\cdots\!41}{45\!\cdots\!00}a^{5}+\frac{67\!\cdots\!79}{45\!\cdots\!00}a^{4}-\frac{27\!\cdots\!21}{90\!\cdots\!60}a^{3}-\frac{20\!\cdots\!03}{76\!\cdots\!00}a^{2}+\frac{32\!\cdots\!51}{22\!\cdots\!90}a+\frac{22\!\cdots\!67}{90\!\cdots\!60}$, $\frac{35\!\cdots\!82}{95\!\cdots\!75}a^{15}-\frac{32\!\cdots\!77}{38\!\cdots\!00}a^{14}-\frac{27\!\cdots\!87}{38\!\cdots\!00}a^{13}+\frac{98\!\cdots\!67}{38\!\cdots\!00}a^{12}+\frac{60\!\cdots\!51}{95\!\cdots\!75}a^{11}-\frac{83\!\cdots\!21}{19\!\cdots\!50}a^{10}-\frac{54\!\cdots\!79}{95\!\cdots\!75}a^{9}+\frac{28\!\cdots\!91}{38\!\cdots\!00}a^{8}-\frac{42\!\cdots\!57}{38\!\cdots\!00}a^{7}-\frac{46\!\cdots\!51}{76\!\cdots\!02}a^{6}+\frac{25\!\cdots\!03}{38\!\cdots\!00}a^{5}+\frac{44\!\cdots\!11}{19\!\cdots\!50}a^{4}-\frac{58\!\cdots\!41}{76\!\cdots\!20}a^{3}-\frac{96\!\cdots\!13}{19\!\cdots\!50}a^{2}-\frac{33\!\cdots\!13}{76\!\cdots\!20}a-\frac{13\!\cdots\!49}{76\!\cdots\!20}$, $\frac{49\!\cdots\!77}{22\!\cdots\!00}a^{15}-\frac{42\!\cdots\!93}{22\!\cdots\!00}a^{14}-\frac{80\!\cdots\!83}{22\!\cdots\!00}a^{13}+\frac{42\!\cdots\!83}{14\!\cdots\!75}a^{12}+\frac{32\!\cdots\!43}{11\!\cdots\!00}a^{11}-\frac{55\!\cdots\!39}{11\!\cdots\!00}a^{10}+\frac{93\!\cdots\!31}{22\!\cdots\!00}a^{9}+\frac{85\!\cdots\!69}{22\!\cdots\!00}a^{8}-\frac{89\!\cdots\!19}{11\!\cdots\!00}a^{7}-\frac{17\!\cdots\!21}{91\!\cdots\!16}a^{6}+\frac{69\!\cdots\!51}{11\!\cdots\!00}a^{5}+\frac{18\!\cdots\!23}{22\!\cdots\!00}a^{4}-\frac{15\!\cdots\!99}{28\!\cdots\!55}a^{3}-\frac{37\!\cdots\!59}{22\!\cdots\!00}a^{2}-\frac{44\!\cdots\!67}{45\!\cdots\!80}a-\frac{12\!\cdots\!23}{22\!\cdots\!40}$, $\frac{98\!\cdots\!60}{55\!\cdots\!21}a^{15}-\frac{12\!\cdots\!60}{55\!\cdots\!21}a^{14}-\frac{17\!\cdots\!20}{55\!\cdots\!21}a^{13}+\frac{22\!\cdots\!00}{55\!\cdots\!21}a^{12}+\frac{14\!\cdots\!60}{55\!\cdots\!21}a^{11}-\frac{30\!\cdots\!50}{55\!\cdots\!21}a^{10}+\frac{10\!\cdots\!40}{55\!\cdots\!21}a^{9}+\frac{19\!\cdots\!20}{55\!\cdots\!21}a^{8}-\frac{50\!\cdots\!40}{55\!\cdots\!21}a^{7}-\frac{83\!\cdots\!70}{55\!\cdots\!21}a^{6}+\frac{30\!\cdots\!40}{55\!\cdots\!21}a^{5}+\frac{34\!\cdots\!60}{55\!\cdots\!21}a^{4}-\frac{65\!\cdots\!60}{55\!\cdots\!21}a^{3}-\frac{11\!\cdots\!70}{55\!\cdots\!21}a^{2}-\frac{68\!\cdots\!00}{55\!\cdots\!21}a-\frac{31\!\cdots\!27}{55\!\cdots\!21}$, $\frac{12\!\cdots\!03}{22\!\cdots\!00}a^{15}-\frac{39\!\cdots\!13}{90\!\cdots\!00}a^{14}-\frac{84\!\cdots\!93}{90\!\cdots\!00}a^{13}+\frac{72\!\cdots\!53}{90\!\cdots\!00}a^{12}+\frac{32\!\cdots\!13}{45\!\cdots\!00}a^{11}-\frac{47\!\cdots\!89}{45\!\cdots\!00}a^{10}+\frac{15\!\cdots\!63}{45\!\cdots\!00}a^{9}+\frac{90\!\cdots\!69}{90\!\cdots\!00}a^{8}-\frac{16\!\cdots\!23}{90\!\cdots\!00}a^{7}-\frac{12\!\cdots\!11}{22\!\cdots\!90}a^{6}+\frac{12\!\cdots\!17}{90\!\cdots\!00}a^{5}+\frac{57\!\cdots\!37}{22\!\cdots\!00}a^{4}-\frac{30\!\cdots\!87}{18\!\cdots\!20}a^{3}-\frac{35\!\cdots\!93}{76\!\cdots\!00}a^{2}-\frac{51\!\cdots\!37}{18\!\cdots\!20}a-\frac{27\!\cdots\!31}{18\!\cdots\!20}$, $\frac{29\!\cdots\!79}{90\!\cdots\!60}a^{15}+\frac{87\!\cdots\!37}{45\!\cdots\!80}a^{14}+\frac{39\!\cdots\!77}{45\!\cdots\!80}a^{13}-\frac{34\!\cdots\!19}{90\!\cdots\!60}a^{12}-\frac{20\!\cdots\!56}{11\!\cdots\!45}a^{11}+\frac{28\!\cdots\!58}{11\!\cdots\!45}a^{10}+\frac{10\!\cdots\!07}{90\!\cdots\!60}a^{9}-\frac{44\!\cdots\!91}{45\!\cdots\!80}a^{8}+\frac{83\!\cdots\!19}{90\!\cdots\!60}a^{7}+\frac{24\!\cdots\!99}{18\!\cdots\!72}a^{6}-\frac{54\!\cdots\!61}{90\!\cdots\!60}a^{5}-\frac{72\!\cdots\!49}{90\!\cdots\!60}a^{4}+\frac{14\!\cdots\!97}{18\!\cdots\!72}a^{3}+\frac{20\!\cdots\!33}{15\!\cdots\!40}a^{2}+\frac{13\!\cdots\!93}{90\!\cdots\!36}a-\frac{42\!\cdots\!29}{18\!\cdots\!72}$ (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)]
| |
Regulator: | \( 12961354.7506 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12961354.7506 \cdot 2}{2\cdot\sqrt{220346584964513535947265625}}\cr\approx \mathstrut & 2.12097809007 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:C_4$ (as 16T33):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3:C_4$ |
Character table for $C_2^3:C_4$ |
Intermediate fields
\(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), 4.0.8405.1 x2, 4.0.1025.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 8.4.14844075753125.1 x2, 8.0.2968815150625.1 x2, 8.0.1766100625.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(41\) | 41.8.6.1 | $x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189402 x^{4} + 1339576 x^{3} + 665040 x^{2} + 8919664 x + 84075609$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
41.8.6.1 | $x^{8} + 152 x^{7} + 8688 x^{6} + 222224 x^{5} + 2189402 x^{4} + 1339576 x^{3} + 665040 x^{2} + 8919664 x + 84075609$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |