Normalized defining polynomial
\( x^{28} - 9 x^{27} + 53 x^{26} - 226 x^{25} + 695 x^{24} - 1455 x^{23} + 1352 x^{22} + 4587 x^{21} - 26437 x^{20} + 75511 x^{19} - 148243 x^{18} + 166438 x^{17} - 8850 x^{16} + \cdots - 465059 \)
Invariants
Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2657211910834657108824251865653514862060546875\) \(\medspace = -\,3^{14}\cdot 5^{21}\cdot 71^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.91\) | 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: | $3^{1/2}5^{3/4}71^{1/2}\approx 48.79971717169349$ | ||
Ramified primes: | \(3\), \(5\), \(71\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-355}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7}a^{18}-\frac{1}{7}a^{17}-\frac{3}{7}a^{14}+\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{19}-\frac{1}{7}a^{17}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{20}-\frac{1}{7}a^{17}-\frac{3}{7}a^{16}-\frac{2}{7}a^{15}-\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{3}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}+\frac{3}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{21}+\frac{3}{7}a^{17}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}-\frac{3}{7}a^{13}-\frac{2}{7}a^{12}-\frac{3}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{22}+\frac{1}{7}a^{17}-\frac{1}{7}a^{16}-\frac{1}{7}a^{15}-\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{91}a^{23}+\frac{2}{91}a^{21}-\frac{4}{91}a^{20}+\frac{4}{91}a^{19}-\frac{6}{91}a^{18}-\frac{9}{91}a^{17}-\frac{5}{13}a^{16}+\frac{5}{13}a^{15}-\frac{32}{91}a^{14}+\frac{2}{91}a^{13}-\frac{34}{91}a^{12}+\frac{17}{91}a^{11}+\frac{1}{7}a^{10}+\frac{22}{91}a^{9}+\frac{27}{91}a^{8}-\frac{18}{91}a^{7}+\frac{25}{91}a^{6}-\frac{27}{91}a^{5}-\frac{3}{7}a^{4}+\frac{20}{91}a^{3}-\frac{4}{91}a^{2}-\frac{5}{91}a-\frac{4}{13}$, $\frac{1}{71071}a^{24}-\frac{290}{71071}a^{23}-\frac{334}{10153}a^{22}-\frac{584}{71071}a^{21}+\frac{1944}{71071}a^{20}-\frac{106}{6461}a^{19}+\frac{2004}{71071}a^{18}+\frac{32397}{71071}a^{17}+\frac{4523}{10153}a^{16}-\frac{27875}{71071}a^{15}-\frac{1217}{5467}a^{14}+\frac{18561}{71071}a^{13}-\frac{4491}{10153}a^{12}-\frac{33673}{71071}a^{11}-\frac{3098}{71071}a^{10}+\frac{32920}{71071}a^{9}+\frac{12484}{71071}a^{8}-\frac{18012}{71071}a^{7}+\frac{11716}{71071}a^{6}-\frac{4152}{10153}a^{5}-\frac{17660}{71071}a^{4}+\frac{3123}{6461}a^{3}-\frac{1390}{6461}a^{2}-\frac{29908}{71071}a-\frac{4157}{10153}$, $\frac{1}{71071}a^{25}+\frac{23}{6461}a^{23}+\frac{1647}{71071}a^{22}-\frac{4187}{71071}a^{21}+\frac{2617}{71071}a^{20}+\frac{475}{71071}a^{19}+\frac{2034}{71071}a^{18}+\frac{1502}{6461}a^{17}-\frac{166}{923}a^{16}-\frac{19345}{71071}a^{15}+\frac{27437}{71071}a^{14}-\frac{18903}{71071}a^{13}+\frac{14632}{71071}a^{12}-\frac{9673}{71071}a^{11}+\frac{27964}{71071}a^{10}-\frac{16557}{71071}a^{9}+\frac{13653}{71071}a^{8}+\frac{20155}{71071}a^{7}+\frac{12619}{71071}a^{6}-\frac{34850}{71071}a^{5}-\frac{394}{71071}a^{4}+\frac{450}{6461}a^{3}+\frac{250}{10153}a^{2}+\frac{1089}{6461}a+\frac{1115}{10153}$, $\frac{1}{600478879}a^{26}+\frac{809}{600478879}a^{25}+\frac{729}{600478879}a^{24}+\frac{869590}{600478879}a^{23}+\frac{4438996}{600478879}a^{22}-\frac{957137}{85782697}a^{21}-\frac{32895524}{600478879}a^{20}-\frac{3422414}{54588989}a^{19}+\frac{284860}{4199153}a^{18}+\frac{154790400}{600478879}a^{17}+\frac{8899335}{85782697}a^{16}-\frac{14399713}{600478879}a^{15}+\frac{20378690}{46190683}a^{14}+\frac{198118223}{600478879}a^{13}+\frac{12620827}{54588989}a^{12}+\frac{297909452}{600478879}a^{11}-\frac{10554207}{54588989}a^{10}-\frac{144489837}{600478879}a^{9}+\frac{108597248}{600478879}a^{8}-\frac{71089838}{600478879}a^{7}+\frac{85248803}{600478879}a^{6}-\frac{150015035}{600478879}a^{5}-\frac{13562937}{600478879}a^{4}-\frac{97919436}{600478879}a^{3}-\frac{282181388}{600478879}a^{2}+\frac{3041856}{12254671}a-\frac{5485143}{12254671}$, $\frac{1}{70\!\cdots\!27}a^{27}+\frac{53\!\cdots\!61}{10\!\cdots\!61}a^{26}-\frac{41\!\cdots\!09}{64\!\cdots\!57}a^{25}-\frac{30\!\cdots\!09}{70\!\cdots\!27}a^{24}-\frac{36\!\cdots\!09}{70\!\cdots\!27}a^{23}+\frac{33\!\cdots\!37}{27\!\cdots\!61}a^{22}-\frac{28\!\cdots\!03}{70\!\cdots\!27}a^{21}-\frac{34\!\cdots\!55}{70\!\cdots\!27}a^{20}+\frac{93\!\cdots\!60}{70\!\cdots\!27}a^{19}-\frac{68\!\cdots\!40}{54\!\cdots\!79}a^{18}-\frac{39\!\cdots\!26}{70\!\cdots\!27}a^{17}-\frac{19\!\cdots\!11}{70\!\cdots\!27}a^{16}-\frac{53\!\cdots\!30}{67\!\cdots\!89}a^{15}+\frac{23\!\cdots\!02}{11\!\cdots\!71}a^{14}-\frac{19\!\cdots\!44}{10\!\cdots\!61}a^{13}-\frac{23\!\cdots\!27}{70\!\cdots\!27}a^{12}+\frac{16\!\cdots\!38}{77\!\cdots\!97}a^{11}+\frac{22\!\cdots\!95}{70\!\cdots\!27}a^{10}-\frac{11\!\cdots\!12}{24\!\cdots\!57}a^{9}+\frac{13\!\cdots\!92}{70\!\cdots\!27}a^{8}-\frac{10\!\cdots\!98}{10\!\cdots\!61}a^{7}-\frac{28\!\cdots\!94}{10\!\cdots\!61}a^{6}-\frac{29\!\cdots\!22}{64\!\cdots\!57}a^{5}+\frac{39\!\cdots\!91}{70\!\cdots\!27}a^{4}-\frac{17\!\cdots\!44}{70\!\cdots\!27}a^{3}-\frac{12\!\cdots\!10}{70\!\cdots\!27}a^{2}+\frac{78\!\cdots\!07}{59\!\cdots\!33}a-\frac{25\!\cdots\!17}{14\!\cdots\!23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $14$ | 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: | not computed | 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: | not computed | 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 $
Galois group
A solvable group of order 56 |
The 17 conjugacy class representatives for $D_{28}$ |
Character table for $D_{28}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.2.79875.1, 7.1.357911.1, 14.2.10007834681328125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 28 sibling: | deg 28 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $28$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{14}$ | ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | ${\href{/padicField/19.7.0.1}{7} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{14}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/47.2.0.1}{2} }^{14}$ | ${\href{/padicField/53.2.0.1}{2} }^{14}$ | ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $28$ | $2$ | $14$ | $14$ | |||
\(5\) | Deg $28$ | $4$ | $7$ | $21$ | |||
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.355.2t1.a.a | $1$ | $ 5 \cdot 71 $ | \(\Q(\sqrt{-355}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.15975.4t3.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 4.0.5671125.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.1775.14t3.a.c | $2$ | $ 5^{2} \cdot 71 $ | 14.0.710556262374296875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.1775.14t3.a.b | $2$ | $ 5^{2} \cdot 71 $ | 14.0.710556262374296875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.1775.14t3.a.a | $2$ | $ 5^{2} \cdot 71 $ | 14.0.710556262374296875.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.15975.28t10.a.d | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.15975.28t10.a.c | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.15975.28t10.a.e | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.15975.28t10.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.15975.28t10.a.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
* | 2.15975.28t10.a.f | $2$ | $ 3^{2} \cdot 5^{2} \cdot 71 $ | 28.2.2657211910834657108824251865653514862060546875.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |