Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1)
gp: K = bnfinit(y^22 - 2*y^20 - 56*y^18 + 245*y^16 + 126*y^14 - 1903*y^12 + 1997*y^10 + 1141*y^8 - 1026*y^6 - 464*y^4 - 41*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1)
\( x^{22} - 2 x^{20} - 56 x^{18} + 245 x^{16} + 126 x^{14} - 1903 x^{12} + 1997 x^{10} + 1141 x^{8} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(56501459388151144478039723653407440896\)
\(\medspace = 2^{22}\cdot 1297^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.00\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(1297\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{10}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{11}-\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{122856150065}a^{20}+\frac{1535557137}{122856150065}a^{18}+\frac{9161641685}{24571230013}a^{16}-\frac{8137363043}{24571230013}a^{14}-\frac{6931598054}{122856150065}a^{12}+\frac{54014683241}{122856150065}a^{10}+\frac{2707745619}{122856150065}a^{8}+\frac{5255859219}{122856150065}a^{6}-\frac{5466880887}{11168740915}a^{4}+\frac{34994977279}{122856150065}a^{2}+\frac{25783075507}{122856150065}$, $\frac{1}{122856150065}a^{21}+\frac{1535557137}{122856150065}a^{19}+\frac{9161641685}{24571230013}a^{17}-\frac{8137363043}{24571230013}a^{15}-\frac{6931598054}{122856150065}a^{13}+\frac{54014683241}{122856150065}a^{11}+\frac{2707745619}{122856150065}a^{9}+\frac{5255859219}{122856150065}a^{7}-\frac{5466880887}{11168740915}a^{5}+\frac{34994977279}{122856150065}a^{3}+\frac{25783075507}{122856150065}a$
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
Trivial group, which has order $1$ (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: | $15$ | 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: |
$a^{21}-2a^{19}-56a^{17}+245a^{15}+126a^{13}-1903a^{11}+1997a^{9}+1141a^{7}-1026a^{5}-464a^{3}-41a$, $\frac{33544520799}{122856150065}a^{20}-\frac{13603768393}{24571230013}a^{18}-\frac{375076652485}{24571230013}a^{16}+\frac{1654390896832}{24571230013}a^{14}+\frac{3935054557624}{122856150065}a^{12}-\frac{63847665178169}{122856150065}a^{10}+\frac{69167353635399}{122856150065}a^{8}+\frac{35502970388093}{122856150065}a^{6}-\frac{3216814305619}{11168740915}a^{4}-\frac{14348096043926}{122856150065}a^{2}-\frac{177152850273}{24571230013}$, $\frac{74023279261}{122856150065}a^{20}-\frac{154437210203}{122856150065}a^{18}-\frac{826177308159}{24571230013}a^{16}+\frac{3698265263846}{24571230013}a^{14}+\frac{7672209785481}{122856150065}a^{12}-\frac{141323137224869}{122856150065}a^{10}+\frac{160198706196299}{122856150065}a^{8}+\frac{69115511997459}{122856150065}a^{6}-\frac{7297285018987}{11168740915}a^{4}-\frac{27308705491556}{122856150065}a^{2}-\frac{1030847839863}{122856150065}$, $\frac{29954979611}{122856150065}a^{20}-\frac{62320073381}{122856150065}a^{18}-\frac{335326857561}{24571230013}a^{16}+\frac{1493539596588}{24571230013}a^{14}+\frac{3386571478406}{122856150065}a^{12}-\frac{57556014534037}{122856150065}a^{10}+\frac{62901099245957}{122856150065}a^{8}+\frac{31770791649991}{122856150065}a^{6}-\frac{2922742197798}{11168740915}a^{4}-\frac{12952080492068}{122856150065}a^{2}-\frac{725489442301}{122856150065}$, $\frac{160274809064}{122856150065}a^{21}-\frac{324139159316}{122856150065}a^{19}-\frac{1793938107800}{24571230013}a^{17}+\frac{7893215439060}{24571230013}a^{15}+\frac{19390369440844}{122856150065}a^{13}-\frac{61110288945602}{24571230013}a^{11}+\frac{65272088868988}{24571230013}a^{9}+\frac{176607302752582}{122856150065}a^{7}-\frac{15288557530706}{11168740915}a^{5}-\frac{14226543643933}{24571230013}a^{3}-\frac{5052395469701}{122856150065}a$, $\frac{7108338071}{24571230013}a^{20}-\frac{73876058122}{122856150065}a^{18}-\frac{396879774556}{24571230013}a^{16}+\frac{1772822827294}{24571230013}a^{14}+\frac{755636558112}{24571230013}a^{12}-\frac{67930119105232}{122856150065}a^{10}+\frac{76262664963362}{122856150065}a^{8}+\frac{34599191477733}{122856150065}a^{6}-\frac{3506935142769}{11168740915}a^{4}-\frac{14027675298363}{122856150065}a^{2}-\frac{635467877802}{122856150065}$, $\frac{86381242551}{122856150065}a^{21}-\frac{177894138316}{122856150065}a^{19}-\frac{964805331180}{24571230013}a^{17}+\frac{4290442721187}{24571230013}a^{15}+\frac{9466576361051}{122856150065}a^{13}-\frac{164665439037772}{122856150065}a^{11}+\frac{183167270422437}{122856150065}a^{9}+\frac{85249844629911}{122856150065}a^{7}-\frac{8470883373908}{11168740915}a^{5}-\frac{33199598002778}{122856150065}a^{3}-\frac{1864987787291}{122856150065}a$, $\frac{3025072135}{24571230013}a^{21}-\frac{23826389823}{122856150065}a^{19}-\frac{172380091403}{24571230013}a^{17}+\frac{670264271023}{24571230013}a^{15}+\frac{718696358665}{24571230013}a^{13}-\frac{28539679178073}{122856150065}a^{11}+\frac{17916160812038}{122856150065}a^{9}+\frac{34216051975367}{122856150065}a^{7}-\frac{1247293225581}{11168740915}a^{5}-\frac{14505752580597}{122856150065}a^{3}-\frac{1168050647838}{122856150065}a$, $\frac{47959946241}{122856150065}a^{20}-\frac{95082271699}{122856150065}a^{18}-\frac{537816252794}{24571230013}a^{16}+\frac{2340711880752}{24571230013}a^{14}+\frac{6334223421111}{122856150065}a^{12}-\frac{18283101364465}{24571230013}a^{10}+\frac{18772186551707}{24571230013}a^{8}+\frac{58318748388768}{122856150065}a^{6}-\frac{4566991827259}{11168740915}a^{4}-\frac{4662975577708}{24571230013}a^{2}-\frac{1315817699684}{122856150065}$, $\frac{452937444059}{122856150065}a^{21}-\frac{930370499868}{122856150065}a^{19}-\frac{5063060945659}{24571230013}a^{17}+\frac{22467319896344}{24571230013}a^{15}+\frac{51051419360979}{122856150065}a^{13}-\frac{864750540668947}{122856150065}a^{11}+\frac{950831703799622}{122856150065}a^{9}+\frac{93180428758724}{24571230013}a^{7}-\frac{8895876889192}{2233748183}a^{5}-\frac{183942935983358}{122856150065}a^{3}-\frac{8948423682108}{122856150065}a$, $\frac{31439440242}{24571230013}a^{21}+\frac{92028245891}{122856150065}a^{20}-\frac{323590532608}{122856150065}a^{19}-\frac{187199408521}{122856150065}a^{18}-\frac{1756693451589}{24571230013}a^{17}-\frac{1028666703392}{24571230013}a^{16}+\frac{7805492133470}{24571230013}a^{15}+\frac{4545437548010}{24571230013}a^{14}+\frac{3498112721327}{24571230013}a^{13}+\frac{10619861728186}{122856150065}a^{12}-\frac{300092305680278}{122856150065}a^{11}-\frac{175182629513157}{122856150065}a^{10}+\frac{331631837670718}{122856150065}a^{9}+\frac{191158539708877}{122856150065}a^{8}+\frac{159374633770657}{122856150065}a^{7}+\frac{95663468736236}{122856150065}a^{6}-\frac{15457682209481}{11168740915}a^{5}-\frac{8941534648448}{11168740915}a^{4}-\frac{63120689018067}{122856150065}a^{3}-\frac{37671502888028}{122856150065}a^{2}-\frac{3164531111853}{122856150065}a-\frac{1866888397376}{122856150065}$, $\frac{213154389896}{122856150065}a^{21}-\frac{34431907569}{122856150065}a^{20}-\frac{433151221427}{122856150065}a^{19}+\frac{70435478273}{122856150065}a^{18}-\frac{2383560352662}{24571230013}a^{17}+\frac{385288554483}{24571230013}a^{16}+\frac{10521884932778}{24571230013}a^{15}-\frac{1705890033226}{24571230013}a^{14}+\frac{24902811392271}{122856150065}a^{13}-\frac{4046812500989}{122856150065}a^{12}-\frac{405962934525913}{122856150065}a^{11}+\frac{66169600020032}{122856150065}a^{10}+\frac{440698552602848}{122856150065}a^{9}-\frac{71284881808707}{122856150065}a^{8}+\frac{44974763464447}{24571230013}a^{7}-\frac{7910432710429}{24571230013}a^{6}-\frac{4146289588530}{2233748183}a^{5}+\frac{693036184261}{2233748183}a^{4}-\frac{88921795667677}{122856150065}a^{3}+\frac{15940924823513}{122856150065}a^{2}-\frac{4517515548662}{122856150065}a+\frac{735925770253}{122856150065}$, $\frac{12547703244}{11168740915}a^{21}+\frac{3226838424}{24571230013}a^{20}-\frac{5177696938}{2233748183}a^{19}-\frac{30964505157}{122856150065}a^{18}-\frac{140209322806}{2233748183}a^{17}-\frac{180297131051}{24571230013}a^{16}+\frac{623611278605}{2233748183}a^{15}+\frac{776610903241}{24571230013}a^{14}+\frac{1383654128234}{11168740915}a^{13}+\frac{421384560185}{24571230013}a^{12}-\frac{23945533810109}{11168740915}a^{11}-\frac{30050072405437}{122856150065}a^{10}+\frac{26550487254859}{11168740915}a^{9}+\frac{31355337848772}{122856150065}a^{8}+\frac{12490395331038}{11168740915}a^{7}+\frac{16651858520903}{122856150065}a^{6}-\frac{13490465576944}{11168740915}a^{5}-\frac{1384380118659}{11168740915}a^{4}-\frac{4834288760061}{11168740915}a^{3}-\frac{5426516745368}{122856150065}a^{2}-\frac{36908927998}{2233748183}a-\frac{233046846842}{122856150065}$, $\frac{187816637981}{122856150065}a^{21}+\frac{20279618848}{122856150065}a^{20}-\frac{399142245636}{122856150065}a^{19}-\frac{13419515842}{24571230013}a^{18}-\frac{2104247692535}{24571230013}a^{17}-\frac{233930692137}{24571230013}a^{16}+\frac{9450279638116}{24571230013}a^{15}+\frac{1264210811422}{24571230013}a^{14}+\frac{20461165800931}{122856150065}a^{13}+\frac{468011104768}{122856150065}a^{12}-\frac{363559443533672}{122856150065}a^{11}-\frac{47636092827753}{122856150065}a^{10}+\frac{401334229122367}{122856150065}a^{9}+\frac{59382693578518}{122856150065}a^{8}+\frac{197898084830531}{122856150065}a^{7}+\frac{24271541291896}{122856150065}a^{6}-\frac{18732004378068}{11168740915}a^{5}-\frac{2742161309788}{11168740915}a^{4}-\frac{79065638804613}{122856150065}a^{3}-\frac{10369628747587}{122856150065}a^{2}-\frac{4297506508021}{122856150065}a-\frac{100444429710}{24571230013}$, $\frac{208927633228}{122856150065}a^{21}-\frac{663586910}{2233748183}a^{20}-\frac{401181194808}{122856150065}a^{19}+\frac{2339386212}{11168740915}a^{18}-\frac{2329984599064}{24571230013}a^{17}+\frac{36298389113}{2233748183}a^{16}+\frac{10067524434882}{24571230013}a^{15}-\frac{117442634952}{2233748183}a^{14}+\frac{26051159511313}{122856150065}a^{13}-\frac{159978538202}{2233748183}a^{12}-\frac{388057361400906}{122856150065}a^{11}+\frac{4717074769522}{11168740915}a^{10}+\frac{415536570344861}{122856150065}a^{9}-\frac{3172700028352}{11168740915}a^{8}+\frac{205479892495463}{122856150065}a^{7}-\frac{2842507720813}{11168740915}a^{6}-\frac{18765049311899}{11168740915}a^{5}+\frac{967893926899}{11168740915}a^{4}-\frac{73726912434814}{122856150065}a^{3}+\frac{453639953083}{11168740915}a^{2}-\frac{2665549031298}{122856150065}a+\frac{14555188662}{11168740915}$
| 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: | \( 101394162831 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 101394162831 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.424944577871789 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1)
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^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1, 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^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1);
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^22 - 2*x^20 - 56*x^18 + 245*x^16 + 126*x^14 - 1903*x^12 + 1997*x^10 + 1141*x^8 - 1026*x^6 - 464*x^4 - 41*x^2 - 1);
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
$C_2^{10}.D_{11}$ (as 22T30):
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)
| A solvable group of order 22528 |
| The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ |
| Character table for $C_2^{10}.D_{11}$ |
Intermediate fields
| 11.11.3670285774226257.1 |
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]
Sibling fields
| Degree 22 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.14.73282392826432034388017521578469450842112.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{10}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(1297\)
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |