Properties

Label 15680u2
Conductor 1568015680
Discriminant 1.851×1016-1.851\times 10^{16}
j-invariant 5452947409250 -\frac{5452947409}{250}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x3+x21539841x+734980959y^2=x^3+x^2-1539841x+734980959 Copy content Toggle raw display (homogenize, simplify)
y2z=x3+x2z1539841xz2+734980959z3y^2z=x^3+x^2z-1539841xz^2+734980959z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3124727148x+536175300528y^2=x^3-124727148x+536175300528 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 0, -1539841, 734980959])
 
gp: E = ellinit([0, 1, 0, -1539841, 734980959])
 
magma: E := EllipticCurve([0, 1, 0, -1539841, 734980959]);
 
oscar: E = elliptic_curve([0, 1, 0, -1539841, 734980959])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);
 

Invariants

Conductor: NN  =  15680 15680  = 265722^{6} \cdot 5 \cdot 7^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  18512297918464000-18512297918464000 = 121953710-1 \cdot 2^{19} \cdot 5^{3} \cdot 7^{10}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  5452947409250 -\frac{5452947409}{250}  = 1215372133373-1 \cdot 2^{-1} \cdot 5^{-3} \cdot 7^{2} \cdot 13^{3} \cdot 37^{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}} ≈ 2.19706445816040945185129891172.1970644581604094518512989117
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.46424810355893626652900989002-0.46424810355893626652900989002
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.98622144374838380.9862214437483838
Szpiro ratio: σm\sigma_{m} ≈ 5.6267553874632195.626755387463219

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.364489215884280117086408126130.36448921588428011708640812613
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 = 2 2  = 211 2\cdot1\cdot1
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) ≈ 0.728978431768560234172816252260.72897843176856023417281625226
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    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

0.728978432L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.3644891.0000002120.728978432\displaystyle 0.728978432 \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 0.364489 \cdot 1.000000 \cdot 2}{1^2} \approx 0.728978432

# 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   15680.2.a.m

q2q3q5+q93q11+5q13+2q156q17q19+O(q20) q - 2 q^{3} - q^{5} + q^{9} - 3 q^{11} + 5 q^{13} + 2 q^{15} - 6 q^{17} - 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: 241920
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 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 22 I9I_{9}^{*} additive 1 6 19 1
55 11 I3I_{3} nonsplit multiplicative 1 1 3 3
77 11 IIII^{*} additive -1 2 10 0

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 except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
33 3B 3.4.0.1

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 = [[4, 3, 9, 7], [835, 6, 834, 7], [419, 714, 777, 461], [218, 147, 175, 547], [599, 0, 0, 839], [337, 126, 651, 379], [1, 6, 0, 1], [3, 4, 8, 11], [631, 126, 693, 379], [1, 0, 6, 1]]
 
GL(2,Integers(840)).subgroup(gens)
 
Gens := [[4, 3, 9, 7], [835, 6, 834, 7], [419, 714, 777, 461], [218, 147, 175, 547], [599, 0, 0, 839], [337, 126, 651, 379], [1, 6, 0, 1], [3, 4, 8, 11], [631, 126, 693, 379], [1, 0, 6, 1]];
 
sub<GL(2,Integers(840))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 840=23357 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 , index 1616, genus 00, and generators

(4397),(83568347),(419714777461),(218147175547),(59900839),(337126651379),(1601),(34811),(631126693379),(1061)\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 835 & 6 \\ 834 & 7 \end{array}\right),\left(\begin{array}{rr} 419 & 714 \\ 777 & 461 \end{array}\right),\left(\begin{array}{rr} 218 & 147 \\ 175 & 547 \end{array}\right),\left(\begin{array}{rr} 599 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 337 & 126 \\ 651 & 379 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 631 & 126 \\ 693 & 379 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \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[840])K:=\Q(E[840]) is a degree-44590694404459069440 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/840Z)\GL_2(\Z/840\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 44 245=572 245 = 5 \cdot 7^{2}
33 good 22 3136=2672 3136 = 2^{6} \cdot 7^{2}
55 nonsplit multiplicative 66 3136=2672 3136 = 2^{6} \cdot 7^{2}
77 additive 2020 320=265 320 = 2^{6} \cdot 5

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3.
Its isogeny class 15680u consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 490e2, its twist by 56-56.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(42)\Q(\sqrt{42}) Z/3Z\Z/3\Z not in database
33 3.1.1960.1 Z/2Z\Z/2\Z not in database
66 6.0.153664000.2 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
66 6.0.25092716544.2 Z/3Z\Z/3\Z not in database
66 6.2.23233996800.1 Z/6Z\Z/6\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/3ZZ/3Z\Z/3\Z \oplus \Z/3\Z not in database
1212 deg 12 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1818 18.6.18980794851677922915110481297408000000000000.2 Z/9Z\Z/9\Z not in database
1818 18.0.3949872260959852271518189879296000000.1 Z/6Z\Z/6\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ord nonsplit add ord ord ord ord ord ord ord ord ord ord ord
λ\lambda-invariant(s) - 4 0 - 0 0 0 0 0 0 0 0 0 0 0
μ\mu-invariant(s) - 0 0 - 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.