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Properties

Label 28.0.205...000.1
Degree $28$
Signature $[0, 14]$
Discriminant $2.051\times 10^{53}$
Root discriminant \(80.17\)
Ramified primes $2,5,29$
Class number $90944$ (GRH)
Class group [4, 28, 812] (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681)
 
gp: K = bnfinit(y^28 + 69*y^26 + 1974*y^24 + 30865*y^22 + 294090*y^20 + 1804843*y^18 + 7322517*y^16 + 19717109*y^14 + 34627149*y^12 + 38013350*y^10 + 24218096*y^8 + 8079074*y^6 + 1320817*y^4 + 92266*y^2 + 1681, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681)
 

\( x^{28} + 69 x^{26} + 1974 x^{24} + 30865 x^{22} + 294090 x^{20} + 1804843 x^{18} + 7322517 x^{16} + \cdots + 1681 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(205102941494858641164670947835018741350400000000000000\) \(\medspace = 2^{28}\cdot 5^{14}\cdot 29^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(80.17\)
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:  $2\cdot 5^{1/2}29^{6/7}\approx 80.1676210131536$
Ramified primes:   \(2\), \(5\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(451,·)$, $\chi_{580}(199,·)$, $\chi_{580}(139,·)$, $\chi_{580}(141,·)$, $\chi_{580}(59,·)$, $\chi_{580}(401,·)$, $\chi_{580}(339,·)$, $\chi_{580}(529,·)$, $\chi_{580}(471,·)$, $\chi_{580}(281,·)$, $\chi_{580}(431,·)$, $\chi_{580}(349,·)$, $\chi_{580}(459,·)$, $\chi_{580}(161,·)$, $\chi_{580}(291,·)$, $\chi_{580}(81,·)$, $\chi_{580}(169,·)$, $\chi_{580}(429,·)$, $\chi_{580}(239,·)$, $\chi_{580}(49,·)$, $\chi_{580}(371,·)$, $\chi_{580}(181,·)$, $\chi_{580}(219,·)$, $\chi_{580}(489,·)$, $\chi_{580}(111,·)$, $\chi_{580}(571,·)$, $\chi_{580}(509,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{41}a^{19}+\frac{5}{41}a^{17}+\frac{16}{41}a^{15}-\frac{15}{41}a^{13}-\frac{4}{41}a^{11}-\frac{3}{41}a^{9}-\frac{14}{41}a^{7}-\frac{3}{41}a^{5}+\frac{1}{41}a^{3}+\frac{16}{41}a$, $\frac{1}{41}a^{20}+\frac{5}{41}a^{18}+\frac{16}{41}a^{16}-\frac{15}{41}a^{14}-\frac{4}{41}a^{12}-\frac{3}{41}a^{10}-\frac{14}{41}a^{8}-\frac{3}{41}a^{6}+\frac{1}{41}a^{4}+\frac{16}{41}a^{2}$, $\frac{1}{41}a^{21}-\frac{9}{41}a^{17}-\frac{13}{41}a^{15}-\frac{11}{41}a^{13}+\frac{17}{41}a^{11}+\frac{1}{41}a^{9}-\frac{15}{41}a^{7}+\frac{16}{41}a^{5}+\frac{11}{41}a^{3}+\frac{2}{41}a$, $\frac{1}{697}a^{22}+\frac{6}{697}a^{20}+\frac{103}{697}a^{18}+\frac{329}{697}a^{16}-\frac{224}{697}a^{14}+\frac{116}{697}a^{12}-\frac{181}{697}a^{10}-\frac{222}{697}a^{8}-\frac{125}{697}a^{6}-\frac{188}{697}a^{4}-\frac{66}{697}a^{2}-\frac{1}{17}$, $\frac{1}{697}a^{23}+\frac{6}{697}a^{21}+\frac{1}{697}a^{19}-\frac{181}{697}a^{17}+\frac{235}{697}a^{15}+\frac{252}{697}a^{13}+\frac{227}{697}a^{11}+\frac{84}{697}a^{9}-\frac{91}{697}a^{7}+\frac{118}{697}a^{5}-\frac{168}{697}a^{3}-\frac{279}{697}a$, $\frac{1}{4980118669}a^{24}+\frac{2783324}{4980118669}a^{22}-\frac{59097396}{4980118669}a^{20}+\frac{260142159}{4980118669}a^{18}-\frac{799373620}{4980118669}a^{16}+\frac{1385484718}{4980118669}a^{14}-\frac{1906305581}{4980118669}a^{12}-\frac{1815298344}{4980118669}a^{10}-\frac{2191529635}{4980118669}a^{8}+\frac{1939393871}{4980118669}a^{6}+\frac{1967003488}{4980118669}a^{4}-\frac{1175056885}{4980118669}a^{2}+\frac{43383412}{121466309}$, $\frac{1}{4980118669}a^{25}+\frac{2783324}{4980118669}a^{23}-\frac{59097396}{4980118669}a^{21}+\frac{17209541}{4980118669}a^{19}-\frac{2014036710}{4980118669}a^{17}+\frac{2478681499}{4980118669}a^{15}+\frac{42382529}{121466309}a^{13}-\frac{843567872}{4980118669}a^{11}-\frac{1462731781}{4980118669}a^{9}+\frac{360331854}{4980118669}a^{7}-\frac{2284317327}{4980118669}a^{5}-\frac{1417989503}{4980118669}a^{3}-\frac{2108201996}{4980118669}a$, $\frac{1}{48\!\cdots\!69}a^{26}+\frac{2581602224}{48\!\cdots\!69}a^{24}-\frac{32\!\cdots\!64}{48\!\cdots\!69}a^{22}-\frac{40\!\cdots\!44}{48\!\cdots\!69}a^{20}-\frac{67\!\cdots\!26}{28\!\cdots\!57}a^{18}-\frac{14\!\cdots\!06}{48\!\cdots\!69}a^{16}+\frac{23\!\cdots\!71}{48\!\cdots\!69}a^{14}+\frac{93\!\cdots\!19}{48\!\cdots\!69}a^{12}+\frac{12\!\cdots\!29}{48\!\cdots\!69}a^{10}-\frac{66\!\cdots\!82}{48\!\cdots\!69}a^{8}+\frac{20\!\cdots\!26}{48\!\cdots\!69}a^{6}-\frac{88\!\cdots\!00}{48\!\cdots\!69}a^{4}-\frac{20\!\cdots\!28}{48\!\cdots\!69}a^{2}+\frac{33\!\cdots\!73}{28\!\cdots\!49}$, $\frac{1}{48\!\cdots\!69}a^{27}+\frac{2581602224}{48\!\cdots\!69}a^{25}-\frac{32\!\cdots\!64}{48\!\cdots\!69}a^{23}-\frac{40\!\cdots\!44}{48\!\cdots\!69}a^{21}+\frac{21\!\cdots\!44}{28\!\cdots\!57}a^{19}+\frac{92\!\cdots\!75}{48\!\cdots\!69}a^{17}+\frac{18\!\cdots\!35}{48\!\cdots\!69}a^{15}-\frac{22\!\cdots\!24}{48\!\cdots\!69}a^{13}+\frac{13\!\cdots\!38}{48\!\cdots\!69}a^{11}+\frac{63\!\cdots\!17}{48\!\cdots\!69}a^{9}+\frac{55\!\cdots\!73}{48\!\cdots\!69}a^{7}+\frac{42\!\cdots\!99}{48\!\cdots\!69}a^{5}-\frac{85\!\cdots\!38}{48\!\cdots\!69}a^{3}+\frac{19\!\cdots\!97}{11\!\cdots\!09}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}\times C_{28}\times C_{812}$, which has order $90944$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{76312508254}{1018332217582181} a^{27} + \frac{4887465691706}{1018332217582181} a^{25} + \frac{125531913681362}{1018332217582181} a^{23} + \frac{1673688288942610}{1018332217582181} a^{21} + \frac{12513121902505317}{1018332217582181} a^{19} + \frac{3033452798317255}{59901895151893} a^{17} + \frac{89691424738823298}{1018332217582181} a^{15} - \frac{132263574798686820}{1018332217582181} a^{13} - \frac{1006293990451061196}{1018332217582181} a^{11} - \frac{2148704059783968070}{1018332217582181} a^{9} - \frac{2239360689609567932}{1018332217582181} a^{7} - \frac{1125749116331006403}{1018332217582181} a^{5} - \frac{219195043656905800}{1018332217582181} a^{3} - \frac{276729386944036}{24837371160541} a \)  (order $4$) Copy content Toggle raw display
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:   $\frac{231144967019221}{11\!\cdots\!09}a^{27}+\frac{14\!\cdots\!96}{11\!\cdots\!09}a^{25}+\frac{39\!\cdots\!77}{11\!\cdots\!09}a^{23}+\frac{53\!\cdots\!52}{11\!\cdots\!09}a^{21}+\frac{42\!\cdots\!27}{11\!\cdots\!09}a^{19}+\frac{11\!\cdots\!18}{69\!\cdots\!77}a^{17}+\frac{53\!\cdots\!17}{11\!\cdots\!09}a^{15}+\frac{64\!\cdots\!95}{11\!\cdots\!09}a^{13}-\frac{35\!\cdots\!76}{11\!\cdots\!09}a^{11}-\frac{20\!\cdots\!46}{11\!\cdots\!09}a^{9}-\frac{23\!\cdots\!95}{11\!\cdots\!09}a^{7}-\frac{11\!\cdots\!69}{11\!\cdots\!09}a^{5}-\frac{22\!\cdots\!65}{11\!\cdots\!09}a^{3}-\frac{17\!\cdots\!92}{11\!\cdots\!09}a$, $\frac{12\!\cdots\!26}{28\!\cdots\!57}a^{26}+\frac{82\!\cdots\!83}{28\!\cdots\!57}a^{24}+\frac{22\!\cdots\!90}{28\!\cdots\!57}a^{22}+\frac{31\!\cdots\!81}{28\!\cdots\!57}a^{20}+\frac{26\!\cdots\!48}{28\!\cdots\!57}a^{18}+\frac{14\!\cdots\!37}{28\!\cdots\!57}a^{16}+\frac{46\!\cdots\!12}{28\!\cdots\!57}a^{14}+\frac{95\!\cdots\!28}{28\!\cdots\!57}a^{12}+\frac{11\!\cdots\!03}{28\!\cdots\!57}a^{10}+\frac{82\!\cdots\!47}{28\!\cdots\!57}a^{8}+\frac{31\!\cdots\!37}{28\!\cdots\!57}a^{6}+\frac{86\!\cdots\!86}{28\!\cdots\!57}a^{4}+\frac{13\!\cdots\!60}{28\!\cdots\!57}a^{2}+\frac{12342972997529}{4913786018551}$, $\frac{40\!\cdots\!16}{48\!\cdots\!69}a^{26}+\frac{26\!\cdots\!28}{48\!\cdots\!69}a^{24}+\frac{72\!\cdots\!86}{48\!\cdots\!69}a^{22}+\frac{10\!\cdots\!50}{48\!\cdots\!69}a^{20}+\frac{93\!\cdots\!20}{48\!\cdots\!69}a^{18}+\frac{30\!\cdots\!40}{28\!\cdots\!57}a^{16}+\frac{17\!\cdots\!52}{48\!\cdots\!69}a^{14}+\frac{39\!\cdots\!52}{48\!\cdots\!69}a^{12}+\frac{54\!\cdots\!40}{48\!\cdots\!69}a^{10}+\frac{43\!\cdots\!19}{48\!\cdots\!69}a^{8}+\frac{16\!\cdots\!30}{48\!\cdots\!69}a^{6}+\frac{21\!\cdots\!82}{48\!\cdots\!69}a^{4}+\frac{19\!\cdots\!55}{48\!\cdots\!69}a^{2}-\frac{61275608384767}{83534362315367}$, $\frac{34\!\cdots\!90}{48\!\cdots\!69}a^{26}+\frac{23\!\cdots\!00}{48\!\cdots\!69}a^{24}+\frac{66\!\cdots\!86}{48\!\cdots\!69}a^{22}+\frac{10\!\cdots\!14}{48\!\cdots\!69}a^{20}+\frac{95\!\cdots\!48}{48\!\cdots\!69}a^{18}+\frac{33\!\cdots\!12}{28\!\cdots\!57}a^{16}+\frac{22\!\cdots\!10}{48\!\cdots\!69}a^{14}+\frac{57\!\cdots\!39}{48\!\cdots\!69}a^{12}+\frac{16\!\cdots\!47}{82\!\cdots\!91}a^{10}+\frac{97\!\cdots\!06}{48\!\cdots\!69}a^{8}+\frac{54\!\cdots\!37}{48\!\cdots\!69}a^{6}+\frac{14\!\cdots\!48}{48\!\cdots\!69}a^{4}+\frac{16\!\cdots\!05}{48\!\cdots\!69}a^{2}+\frac{868328777888259}{83534362315367}$, $\frac{93\!\cdots\!64}{48\!\cdots\!69}a^{26}+\frac{53\!\cdots\!57}{48\!\cdots\!69}a^{24}+\frac{11\!\cdots\!52}{48\!\cdots\!69}a^{22}+\frac{93\!\cdots\!40}{48\!\cdots\!69}a^{20}-\frac{54\!\cdots\!11}{48\!\cdots\!69}a^{18}-\frac{40\!\cdots\!63}{28\!\cdots\!57}a^{16}-\frac{56\!\cdots\!06}{48\!\cdots\!69}a^{14}-\frac{23\!\cdots\!60}{48\!\cdots\!69}a^{12}-\frac{53\!\cdots\!61}{48\!\cdots\!69}a^{10}-\frac{69\!\cdots\!81}{48\!\cdots\!69}a^{8}-\frac{47\!\cdots\!76}{48\!\cdots\!69}a^{6}-\frac{14\!\cdots\!64}{48\!\cdots\!69}a^{4}-\frac{15\!\cdots\!29}{48\!\cdots\!69}a^{2}-\frac{554115530196598}{83534362315367}$, $\frac{35\!\cdots\!78}{48\!\cdots\!69}a^{26}+\frac{25\!\cdots\!49}{48\!\cdots\!69}a^{24}+\frac{74\!\cdots\!50}{48\!\cdots\!69}a^{22}+\frac{12\!\cdots\!36}{48\!\cdots\!69}a^{20}+\frac{12\!\cdots\!31}{48\!\cdots\!69}a^{18}+\frac{47\!\cdots\!65}{28\!\cdots\!57}a^{16}+\frac{34\!\cdots\!32}{48\!\cdots\!69}a^{14}+\frac{96\!\cdots\!89}{48\!\cdots\!69}a^{12}+\frac{17\!\cdots\!70}{48\!\cdots\!69}a^{10}+\frac{18\!\cdots\!54}{48\!\cdots\!69}a^{8}+\frac{11\!\cdots\!07}{48\!\cdots\!69}a^{6}+\frac{30\!\cdots\!41}{48\!\cdots\!69}a^{4}+\frac{31\!\cdots\!56}{48\!\cdots\!69}a^{2}+\frac{11\!\cdots\!74}{83534362315367}$, $\frac{37\!\cdots\!72}{48\!\cdots\!69}a^{26}+\frac{24\!\cdots\!01}{48\!\cdots\!69}a^{24}+\frac{68\!\cdots\!22}{48\!\cdots\!69}a^{22}+\frac{10\!\cdots\!48}{48\!\cdots\!69}a^{20}+\frac{90\!\cdots\!59}{48\!\cdots\!69}a^{18}+\frac{30\!\cdots\!50}{28\!\cdots\!57}a^{16}+\frac{18\!\cdots\!62}{48\!\cdots\!69}a^{14}+\frac{45\!\cdots\!44}{48\!\cdots\!69}a^{12}+\frac{69\!\cdots\!77}{48\!\cdots\!69}a^{10}+\frac{64\!\cdots\!49}{48\!\cdots\!69}a^{8}+\frac{34\!\cdots\!70}{48\!\cdots\!69}a^{6}+\frac{93\!\cdots\!24}{48\!\cdots\!69}a^{4}+\frac{10\!\cdots\!58}{48\!\cdots\!69}a^{2}+\frac{278721074363799}{83534362315367}$, $\frac{16\!\cdots\!03}{48\!\cdots\!69}a^{27}+\frac{11\!\cdots\!77}{48\!\cdots\!69}a^{25}+\frac{32\!\cdots\!81}{48\!\cdots\!69}a^{23}+\frac{50\!\cdots\!55}{48\!\cdots\!69}a^{21}+\frac{47\!\cdots\!14}{48\!\cdots\!69}a^{19}+\frac{29\!\cdots\!95}{48\!\cdots\!69}a^{17}+\frac{69\!\cdots\!34}{28\!\cdots\!57}a^{15}+\frac{31\!\cdots\!19}{48\!\cdots\!69}a^{13}+\frac{55\!\cdots\!11}{48\!\cdots\!69}a^{11}+\frac{61\!\cdots\!97}{48\!\cdots\!69}a^{9}+\frac{39\!\cdots\!08}{48\!\cdots\!69}a^{7}+\frac{12\!\cdots\!05}{48\!\cdots\!69}a^{5}+\frac{11\!\cdots\!80}{48\!\cdots\!69}a^{3}+\frac{56\!\cdots\!28}{11\!\cdots\!09}a$, $\frac{37\!\cdots\!41}{48\!\cdots\!69}a^{26}+\frac{26\!\cdots\!22}{48\!\cdots\!69}a^{24}+\frac{79\!\cdots\!93}{48\!\cdots\!69}a^{22}+\frac{12\!\cdots\!32}{48\!\cdots\!69}a^{20}+\frac{12\!\cdots\!31}{48\!\cdots\!69}a^{18}+\frac{81\!\cdots\!39}{48\!\cdots\!69}a^{16}+\frac{34\!\cdots\!70}{48\!\cdots\!69}a^{14}+\frac{95\!\cdots\!32}{48\!\cdots\!69}a^{12}+\frac{16\!\cdots\!69}{48\!\cdots\!69}a^{10}+\frac{18\!\cdots\!68}{48\!\cdots\!69}a^{8}+\frac{10\!\cdots\!67}{48\!\cdots\!69}a^{6}+\frac{26\!\cdots\!16}{48\!\cdots\!69}a^{4}+\frac{26\!\cdots\!81}{48\!\cdots\!69}a^{2}+\frac{33\!\cdots\!06}{28\!\cdots\!49}$, $\frac{98\!\cdots\!86}{48\!\cdots\!69}a^{27}+\frac{67\!\cdots\!96}{48\!\cdots\!69}a^{25}+\frac{19\!\cdots\!99}{48\!\cdots\!69}a^{23}+\frac{29\!\cdots\!64}{48\!\cdots\!69}a^{21}+\frac{27\!\cdots\!29}{48\!\cdots\!69}a^{19}+\frac{16\!\cdots\!56}{48\!\cdots\!69}a^{17}+\frac{65\!\cdots\!72}{48\!\cdots\!69}a^{15}+\frac{17\!\cdots\!80}{48\!\cdots\!69}a^{13}+\frac{28\!\cdots\!07}{48\!\cdots\!69}a^{11}+\frac{30\!\cdots\!83}{48\!\cdots\!69}a^{9}+\frac{17\!\cdots\!73}{48\!\cdots\!69}a^{7}+\frac{48\!\cdots\!09}{48\!\cdots\!69}a^{5}+\frac{52\!\cdots\!18}{48\!\cdots\!69}a^{3}+\frac{27\!\cdots\!24}{11\!\cdots\!09}a$, $\frac{52\!\cdots\!16}{48\!\cdots\!69}a^{27}+\frac{36\!\cdots\!26}{48\!\cdots\!69}a^{25}+\frac{10\!\cdots\!97}{48\!\cdots\!69}a^{23}+\frac{15\!\cdots\!44}{48\!\cdots\!69}a^{21}+\frac{14\!\cdots\!46}{48\!\cdots\!69}a^{19}+\frac{90\!\cdots\!04}{48\!\cdots\!69}a^{17}+\frac{60\!\cdots\!60}{82\!\cdots\!91}a^{15}+\frac{15\!\cdots\!43}{82\!\cdots\!91}a^{13}+\frac{92\!\cdots\!92}{28\!\cdots\!57}a^{11}+\frac{16\!\cdots\!25}{48\!\cdots\!69}a^{9}+\frac{90\!\cdots\!43}{48\!\cdots\!69}a^{7}+\frac{23\!\cdots\!57}{48\!\cdots\!69}a^{5}+\frac{25\!\cdots\!12}{48\!\cdots\!69}a^{3}+\frac{22\!\cdots\!55}{11\!\cdots\!09}a$, $\frac{15\!\cdots\!82}{48\!\cdots\!69}a^{27}+\frac{62\!\cdots\!07}{28\!\cdots\!57}a^{25}+\frac{29\!\cdots\!37}{48\!\cdots\!69}a^{23}+\frac{45\!\cdots\!75}{48\!\cdots\!69}a^{21}+\frac{41\!\cdots\!89}{48\!\cdots\!69}a^{19}+\frac{24\!\cdots\!12}{48\!\cdots\!69}a^{17}+\frac{93\!\cdots\!53}{48\!\cdots\!69}a^{15}+\frac{23\!\cdots\!56}{48\!\cdots\!69}a^{13}+\frac{38\!\cdots\!07}{48\!\cdots\!69}a^{11}+\frac{38\!\cdots\!91}{48\!\cdots\!69}a^{9}+\frac{22\!\cdots\!29}{48\!\cdots\!69}a^{7}+\frac{64\!\cdots\!76}{48\!\cdots\!69}a^{5}+\frac{48\!\cdots\!81}{28\!\cdots\!57}a^{3}+\frac{79\!\cdots\!49}{11\!\cdots\!09}a$, $\frac{75\!\cdots\!19}{48\!\cdots\!69}a^{27}+\frac{39\!\cdots\!21}{48\!\cdots\!69}a^{25}+\frac{66\!\cdots\!30}{48\!\cdots\!69}a^{23}+\frac{60\!\cdots\!24}{28\!\cdots\!57}a^{21}-\frac{97\!\cdots\!99}{48\!\cdots\!69}a^{19}-\frac{13\!\cdots\!98}{48\!\cdots\!69}a^{17}-\frac{88\!\cdots\!13}{48\!\cdots\!69}a^{15}-\frac{33\!\cdots\!14}{48\!\cdots\!69}a^{13}-\frac{74\!\cdots\!30}{48\!\cdots\!69}a^{11}-\frac{99\!\cdots\!23}{48\!\cdots\!69}a^{9}-\frac{73\!\cdots\!15}{48\!\cdots\!69}a^{7}-\frac{27\!\cdots\!06}{48\!\cdots\!69}a^{5}-\frac{45\!\cdots\!83}{48\!\cdots\!69}a^{3}-\frac{65\!\cdots\!08}{11\!\cdots\!09}a$ Copy content Toggle raw display (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:  \( 34681517373.86067 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 90944}{4\cdot\sqrt{205102941494858641164670947835018741350400000000000000}}\cr\approx \mathstrut & 0.260222211549445 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 69*x^26 + 1974*x^24 + 30865*x^22 + 294090*x^20 + 1804843*x^18 + 7322517*x^16 + 19717109*x^14 + 34627149*x^12 + 38013350*x^10 + 24218096*x^8 + 8079074*x^6 + 1320817*x^4 + 92266*x^2 + 1681);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 7.7.594823321.1, 14.14.27641779937927268828125.1, 14.0.452882922503000372480000000.1, 14.0.5796901408038404767744.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-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 ${\href{/padicField/3.14.0.1}{14} }^{2}$ R ${\href{/padicField/7.14.0.1}{14} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{14}$ ${\href{/padicField/19.14.0.1}{14} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R ${\href{/padicField/31.14.0.1}{14} }^{2}$ ${\href{/padicField/37.14.0.1}{14} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{28}$ ${\href{/padicField/43.14.0.1}{14} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $28$$2$$14$$28$
\(5\) Copy content Toggle raw display 5.14.7.1$x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(29\) Copy content Toggle raw display 29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$