Properties

Label 445280.bf1
Conductor 445280445280
Discriminant 6.599×1023-6.599\times 10^{23}
j-invariant 14605154817835672546564453125 -\frac{14605154817835672}{546564453125}
CM no
Rank 11
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3+x254224496x158597991796y^2=x^3+x^2-54224496x-158597991796 Copy content Toggle raw display (homogenize, simplify)
y2z=x3+x2z54224496xz2158597991796z3y^2z=x^3+x^2z-54224496xz^2-158597991796z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x34392184203x115604759466702y^2=x^3-4392184203x-115604759466702 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 0, -54224496, -158597991796])
 
gp: E = ellinit([0, 1, 0, -54224496, -158597991796])
 
magma: E := EllipticCurve([0, 1, 0, -54224496, -158597991796]);
 
oscar: E = elliptic_curve([0, 1, 0, -54224496, -158597991796])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(10993931354755615/147576600649,1146875380476883166304866/56692584175517893)(10993931354755615/147576600649, 1146875380476883166304866/56692584175517893)31.04346081147209939388739329931.043460811472099393887393299\infty

Integral points

None

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  445280 445280  = 255112232^{5} \cdot 5 \cdot 11^{2} \cdot 23
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  659850439797131000000000-659850439797131000000000 = 12959119234-1 \cdot 2^{9} \cdot 5^{9} \cdot 11^{9} \cdot 23^{4}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  14605154817835672546564453125 -\frac{14605154817835672}{546564453125}  = 123592341222193-1 \cdot 2^{3} \cdot 5^{-9} \cdot 23^{-4} \cdot 122219^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 3.34102198076564381943393816233.3410219807656438194339381623
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.02274014074690692932455638771.0227401407469069293245563877
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.95013552071067140.9501355207106714
Szpiro ratio: σm\sigma_{m} ≈ 5.0053743819451555.005374381945155

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 31.04346081147209939388739329931.043460811472099393887393299
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.0277567004995177171436985850060.027756700499517717143698585006
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 8 8  = 11222 1\cdot1\cdot2\cdot2^{2}
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L'(E,1) ≈ 6.89331235370037038426180880266.8933123537003703842618088026
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

6.893312354L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.02775731.0434618126.893312354\displaystyle 6.893312354 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.027757 \cdot 31.043461 \cdot 8}{1^2} \approx 6.893312354

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 445280.2.a.bf

q+q3q5+q72q94q13q15+5q175q19+O(q20) q + q^{3} - q^{5} + q^{7} - 2 q^{9} - 4 q^{13} - q^{15} + 5 q^{17} - 5 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 52005888
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 11 I0I_0^{*} additive -1 5 9 0
55 11 I9I_{9} nonsplit multiplicative 1 1 9 9
1111 22 IIIIII^{*} additive 1 2 9 0
2323 44 I4I_{4} split multiplicative -1 1 4 4

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell.

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[439, 2, 438, 3], [221, 2, 221, 3], [321, 2, 321, 3], [1, 1, 439, 0], [1, 0, 2, 1], [1, 2, 0, 1], [111, 2, 0, 1], [177, 2, 177, 3]]
 
GL(2,Integers(440)).subgroup(gens)
 
Gens := [[439, 2, 438, 3], [221, 2, 221, 3], [321, 2, 321, 3], [1, 1, 439, 0], [1, 0, 2, 1], [1, 2, 0, 1], [111, 2, 0, 1], [177, 2, 177, 3]];
 
sub<GL(2,Integers(440))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 440=23511 440 = 2^{3} \cdot 5 \cdot 11 , index 22, genus 00, and generators

(43924383),(22122213),(32123213),(114390),(1021),(1201),(111201),(17721773)\left(\begin{array}{rr} 439 & 2 \\ 438 & 3 \end{array}\right),\left(\begin{array}{rr} 221 & 2 \\ 221 & 3 \end{array}\right),\left(\begin{array}{rr} 321 & 2 \\ 321 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 439 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 111 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 177 & 2 \\ 177 & 3 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[440])K:=\Q(E[440]) is a degree-48660480004866048000 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/440Z)\GL_2(\Z/440\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 additive 44 55=511 55 = 5 \cdot 11
33 good 22 89056=2511223 89056 = 2^{5} \cdot 11^{2} \cdot 23
55 nonsplit multiplicative 66 89056=2511223 89056 = 2^{5} \cdot 11^{2} \cdot 23
1111 additive 4242 3680=25523 3680 = 2^{5} \cdot 5 \cdot 23
2323 split multiplicative 2424 19360=255112 19360 = 2^{5} \cdot 5 \cdot 11^{2}

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 445280.bf consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 445280.n1, its twist by 4444.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.