Properties

Label 429429a1
Conductor $429429$
Discriminant $-1.914\times 10^{21}$
j-invariant \( \frac{9208180736}{223810587} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-x^2+892940x+2079273822\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-x^2z+892940xz^2+2079273822z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+1157249808x+97024486450320\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 1, 892940, 2079273822])
 
gp: E = ellinit([0, -1, 1, 892940, 2079273822])
 
magma: E := EllipticCurve([0, -1, 1, 892940, 2079273822]);
 
oscar: E = EllipticCurve([0, -1, 1, 892940, 2079273822])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(-\frac{2811}{4}, \frac{265833}{8}\right)\) Copy content Toggle raw display \(\left(5339, 398755\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.5082586081139974293576231847$$1.8561823238675724473804671958$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\( \left(-975, 16758\right) \), \( \left(-975, -16759\right) \), \( \left(477, 51122\right) \), \( \left(477, -51123\right) \), \( \left(5339, 398755\right) \), \( \left(5339, -398756\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 429429 \)  =  $3 \cdot 7 \cdot 11^{2} \cdot 13^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-1913801325641316474363 $  =  $-1 \cdot 3^{3} \cdot 7^{3} \cdot 11^{7} \cdot 13^{9} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{9208180736}{223810587} \)  =  $2^{12} \cdot 3^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 13^{-3} \cdot 131^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.7638116743101331901893896742\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.28238935918017955013167416444\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $2.4045289642356318590717894303\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.11096572634367079953395725013\dots$
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: $ 16 $  = $ 1\cdot1\cdot2^{2}\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: $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L^{(2)}(E,1)/2! $ ≈ $ 4.2691248484928210552327759860 $
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);
 

BSD formula

$\displaystyle 4.269124848 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.110966 \cdot 2.404529 \cdot 16}{1^2} \approx 4.269124848$

# 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 analyiic 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 429429.2.a.a

\( q - 2 q^{2} - q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - q^{7} + q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{14} + 4 q^{15} - 4 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} + O(q^{20}) \) 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: 47900160
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$7$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$11$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$13$ $4$ $I_{3}^{*}$ Additive 1 2 9 3

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 = [[1, 0, 2, 1], [4369, 2, 4369, 3], [1, 2, 0, 1], [2003, 2, 2003, 3], [3433, 2, 3433, 3], [6005, 2, 6004, 3], [925, 2, 925, 3], [1, 1, 6005, 0]]
 
GL(2,Integers(6006)).subgroup(gens)
 
Gens := [[1, 0, 2, 1], [4369, 2, 4369, 3], [1, 2, 0, 1], [2003, 2, 2003, 3], [3433, 2, 3433, 3], [6005, 2, 6004, 3], [925, 2, 925, 3], [1, 1, 6005, 0]];
 
sub<GL(2,Integers(6006))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 4369 & 2 \\ 4369 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2003 & 2 \\ 2003 & 3 \end{array}\right),\left(\begin{array}{rr} 3433 & 2 \\ 3433 & 3 \end{array}\right),\left(\begin{array}{rr} 6005 & 2 \\ 6004 & 3 \end{array}\right),\left(\begin{array}{rr} 925 & 2 \\ 925 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6005 & 0 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[6006])$ is a degree-$100429391462400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6006\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 429429a consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 3003b1, its twist by $-143$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.