Properties

Label 32.0.113...856.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.133\times 10^{50}$
Root discriminant \(36.66\)
Ramified primes $2,3,7$
Class number $64$ (GRH)
Class group [4, 4, 4] (GRH)
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536)
 
gp: K = bnfinit(y^32 + 31*y^24 + 705*y^16 + 7936*y^8 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536)
 

\( x^{32} + 31x^{24} + 705x^{16} + 7936x^{8} + 65536 \) Copy content Toggle raw display

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(113341328678310292663197722067951895712000278265856\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 7^{16}\) 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:  \(36.66\)
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:  $2^{3}3^{1/2}7^{1/2}\approx 36.66060555964672$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(336=2^{4}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{336}(1,·)$, $\chi_{336}(265,·)$, $\chi_{336}(139,·)$, $\chi_{336}(13,·)$, $\chi_{336}(281,·)$, $\chi_{336}(155,·)$, $\chi_{336}(29,·)$, $\chi_{336}(293,·)$, $\chi_{336}(167,·)$, $\chi_{336}(41,·)$, $\chi_{336}(43,·)$, $\chi_{336}(125,·)$, $\chi_{336}(307,·)$, $\chi_{336}(181,·)$, $\chi_{336}(55,·)$, $\chi_{336}(323,·)$, $\chi_{336}(197,·)$, $\chi_{336}(71,·)$, $\chi_{336}(335,·)$, $\chi_{336}(209,·)$, $\chi_{336}(83,·)$, $\chi_{336}(85,·)$, $\chi_{336}(223,·)$, $\chi_{336}(97,·)$, $\chi_{336}(295,·)$, $\chi_{336}(239,·)$, $\chi_{336}(113,·)$, $\chi_{336}(211,·)$, $\chi_{336}(169,·)$, $\chi_{336}(251,·)$, $\chi_{336}(253,·)$, $\chi_{336}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{9}+\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{1}{12}a^{10}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{19}-\frac{1}{24}a^{11}+\frac{1}{24}a^{3}$, $\frac{1}{48}a^{20}-\frac{1}{48}a^{12}+\frac{1}{48}a^{4}$, $\frac{1}{96}a^{21}-\frac{1}{96}a^{13}+\frac{1}{96}a^{5}$, $\frac{1}{192}a^{22}+\frac{95}{192}a^{14}+\frac{1}{192}a^{6}$, $\frac{1}{384}a^{23}-\frac{97}{384}a^{15}-\frac{191}{384}a^{7}$, $\frac{1}{541440}a^{24}-\frac{11}{256}a^{16}-\frac{85}{256}a^{8}+\frac{736}{2115}$, $\frac{1}{1082880}a^{25}-\frac{11}{512}a^{17}-\frac{85}{512}a^{9}+\frac{368}{2115}a$, $\frac{1}{2165760}a^{26}-\frac{11}{1024}a^{18}-\frac{85}{1024}a^{10}+\frac{184}{2115}a^{2}$, $\frac{1}{4331520}a^{27}-\frac{11}{2048}a^{19}+\frac{939}{2048}a^{11}-\frac{1931}{4230}a^{3}$, $\frac{1}{8663040}a^{28}-\frac{11}{4096}a^{20}+\frac{939}{4096}a^{12}-\frac{1931}{8460}a^{4}$, $\frac{1}{17326080}a^{29}-\frac{11}{8192}a^{21}-\frac{3157}{8192}a^{13}+\frac{6529}{16920}a^{5}$, $\frac{1}{34652160}a^{30}-\frac{11}{16384}a^{22}+\frac{5035}{16384}a^{14}+\frac{6529}{33840}a^{6}$, $\frac{1}{69304320}a^{31}-\frac{11}{32768}a^{23}+\frac{5035}{32768}a^{15}+\frac{6529}{67680}a^{7}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (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:   \( -\frac{7}{270720} a^{31} + \frac{7193}{270720} a^{7} \)  (order $48$) 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:   $\frac{209}{2887680}a^{28}+\frac{15}{4096}a^{20}+\frac{209}{4096}a^{12}+\frac{6479}{11280}a^{4}+1$, $\frac{457}{4331520}a^{27}-\frac{271}{1082880}a^{25}+\frac{23}{6144}a^{19}-\frac{17}{1536}a^{17}+\frac{457}{6144}a^{11}-\frac{271}{1536}a^{9}+\frac{14167}{16920}a^{3}-\frac{8401}{4230}a$, $\frac{1}{27072}a^{30}+\frac{209}{2887680}a^{28}-\frac{341}{2165760}a^{26}+\frac{15}{4096}a^{20}-\frac{11}{3072}a^{18}+\frac{209}{4096}a^{12}-\frac{85}{3072}a^{10}-\frac{4895}{27072}a^{6}+\frac{6479}{11280}a^{4}-\frac{704}{2115}a^{2}+1$, $\frac{1541}{34652160}a^{30}-\frac{287}{17326080}a^{29}+\frac{91}{49152}a^{22}-\frac{1}{24576}a^{21}+\frac{1541}{49152}a^{14}-\frac{287}{24576}a^{13}+\frac{47771}{135360}a^{6}-\frac{8897}{67680}a^{5}$, $\frac{271}{1082880}a^{25}+\frac{17}{1536}a^{17}+\frac{271}{1536}a^{9}+\frac{8401}{4230}a+1$, $\frac{457}{4331520}a^{27}+\frac{23}{6144}a^{19}+\frac{457}{6144}a^{11}+\frac{14167}{16920}a^{3}+1$, $\frac{1}{27072}a^{30}-\frac{271}{1082880}a^{25}-\frac{17}{1536}a^{17}-\frac{271}{1536}a^{9}-\frac{4895}{27072}a^{6}-\frac{8401}{4230}a$, $\frac{43}{2165760}a^{27}-\frac{11}{3072}a^{19}-\frac{85}{3072}a^{11}-\frac{13549}{16920}a^{3}$, $\frac{967}{69304320}a^{31}+\frac{209}{2887680}a^{28}+\frac{713}{4331520}a^{27}+\frac{1}{8460}a^{26}-\frac{31}{180480}a^{25}-\frac{31}{180480}a^{24}+\frac{89}{98304}a^{23}+\frac{15}{4096}a^{20}+\frac{23}{6144}a^{19}-\frac{1}{256}a^{17}-\frac{1}{256}a^{16}+\frac{967}{98304}a^{15}+\frac{209}{4096}a^{12}+\frac{457}{6144}a^{11}-\frac{31}{256}a^{9}-\frac{31}{256}a^{8}+\frac{29977}{270720}a^{7}+\frac{6479}{11280}a^{4}+\frac{736}{2115}a^{3}-\frac{8279}{8460}a^{2}-\frac{961}{705}a-\frac{256}{705}$, $\frac{1}{11280}a^{28}-\frac{17}{433152}a^{27}+\frac{713}{2165760}a^{26}+\frac{1}{4230}a^{25}-\frac{11}{3072}a^{19}+\frac{23}{3072}a^{18}-\frac{85}{3072}a^{11}+\frac{457}{3072}a^{10}-\frac{2639}{11280}a^{4}-\frac{527}{1692}a^{3}+\frac{1472}{2115}a^{2}-\frac{4049}{4230}a$, $\frac{1}{32768}a^{31}+\frac{1}{67680}a^{29}+\frac{209}{2887680}a^{28}-\frac{1}{4230}a^{26}-\frac{271}{1082880}a^{25}-\frac{31}{180480}a^{24}+\frac{31}{32768}a^{23}+\frac{15}{4096}a^{20}-\frac{17}{1536}a^{17}-\frac{1}{256}a^{16}+\frac{705}{32768}a^{15}+\frac{209}{4096}a^{12}-\frac{271}{1536}a^{9}-\frac{31}{256}a^{8}+\frac{31}{128}a^{7}+\frac{8641}{67680}a^{5}+\frac{6479}{11280}a^{4}+\frac{4049}{4230}a^{2}-\frac{8401}{4230}a-\frac{256}{705}$, $\frac{967}{69304320}a^{31}-\frac{1}{11280}a^{29}+\frac{1}{8460}a^{26}+\frac{31}{180480}a^{24}+\frac{89}{98304}a^{23}+\frac{1}{256}a^{16}+\frac{967}{98304}a^{15}+\frac{31}{256}a^{8}+\frac{29977}{270720}a^{7}+\frac{2639}{11280}a^{5}-\frac{8279}{8460}a^{2}+\frac{256}{705}$, $\frac{7}{270720}a^{31}-\frac{1}{67680}a^{29}-\frac{17}{433152}a^{27}+\frac{17}{216576}a^{25}-\frac{11}{3072}a^{19}+\frac{11}{1536}a^{17}-\frac{85}{3072}a^{11}+\frac{85}{1536}a^{9}-\frac{7193}{270720}a^{7}-\frac{8641}{67680}a^{5}-\frac{527}{1692}a^{3}+\frac{527}{846}a$, $\frac{961}{23101440}a^{31}-\frac{1}{27072}a^{30}-\frac{1}{11280}a^{28}-\frac{17}{433152}a^{27}+\frac{1}{8460}a^{26}-\frac{31}{180480}a^{24}+\frac{31}{32768}a^{23}-\frac{11}{3072}a^{19}-\frac{1}{256}a^{16}+\frac{705}{32768}a^{15}-\frac{85}{3072}a^{11}-\frac{31}{256}a^{8}+\frac{62}{705}a^{7}+\frac{4895}{27072}a^{6}+\frac{2639}{11280}a^{4}-\frac{527}{1692}a^{3}+\frac{181}{8460}a^{2}-\frac{256}{705}$, $\frac{2821}{34652160}a^{30}+\frac{1541}{17326080}a^{29}-\frac{1}{11280}a^{28}+\frac{341}{2165760}a^{26}-\frac{31}{180480}a^{25}+\frac{91}{49152}a^{22}+\frac{91}{24576}a^{21}+\frac{11}{3072}a^{18}-\frac{1}{256}a^{17}+\frac{1541}{49152}a^{14}+\frac{1541}{24576}a^{13}+\frac{85}{3072}a^{10}-\frac{31}{256}a^{9}+\frac{364}{2115}a^{6}+\frac{47771}{67680}a^{5}+\frac{2639}{11280}a^{4}+\frac{704}{2115}a^{2}-\frac{961}{705}a-1$ 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:  \( 374180447387.275 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 374180447387.275 \cdot 64}{48\cdot\sqrt{113341328678310292663197722067951895712000278265856}}\cr\approx \mathstrut & 0.276505370236942 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536)
 
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^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536, 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^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536);
 
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^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536);
 
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

$C_2^3\times C_4$ (as 32T34):

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 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{42})\), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{-7})\), 4.4.903168.2, 4.0.903168.5, \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.18432.2, 4.4.18432.1, 4.0.100352.5, 4.4.100352.1, 8.0.12745506816.1, 8.0.12745506816.8, 8.0.49787136.1, 8.0.157351936.1, 8.0.12745506816.9, \(\Q(\zeta_{24})\), 8.0.12745506816.7, 8.0.796594176.2, 8.0.12745506816.3, 8.8.12745506816.1, 8.0.796594176.1, 8.0.12745506816.6, 8.0.12745506816.5, 8.0.12745506816.4, 8.0.12745506816.2, 8.0.3262849744896.7, \(\Q(\zeta_{16})\), 8.0.1358954496.4, 8.0.40282095616.2, 8.8.815712436224.1, 8.0.815712436224.4, 8.0.815712436224.1, 8.8.815712436224.2, 8.0.3262849744896.4, 8.0.3262849744896.5, 8.0.3262849744896.1, 8.0.3262849744896.2, 8.0.815712436224.2, 8.0.815712436224.3, 8.0.10070523904.2, 8.0.10070523904.1, 8.8.3262849744896.1, 8.0.3262849744896.6, 8.8.40282095616.1, 8.0.40282095616.1, 8.0.815712436224.5, 8.0.815712436224.6, 8.0.339738624.2, 8.0.339738624.1, 8.8.3262849744896.2, 8.0.3262849744896.3, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, 16.0.162447943996702457856.1, 16.0.10646188457767892278050816.4, 16.0.10646188457767892278050816.5, 16.0.10646188457767892278050816.8, 16.0.1622647227216566419456.1, 16.0.10646188457767892278050816.7, \(\Q(\zeta_{48})\), 16.0.665386778610493267378176.2, 16.0.665386778610493267378176.1, 16.16.10646188457767892278050816.1, 16.0.10646188457767892278050816.3, 16.0.10646188457767892278050816.9, 16.0.10646188457767892278050816.2, 16.0.10646188457767892278050816.6, 16.0.10646188457767892278050816.1

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 R ${\href{/padicField/5.4.0.1}{4} }^{8}$ R ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{16}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$