Properties

Label 439824r3
Conductor $439824$
Discriminant $-6.390\times 10^{19}$
j-invariant \( \frac{57258048889007}{132611470002} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2+629536x+332903040\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z+629536xz^2+332903040z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+50992389x+242839293354\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 0, 629536, 332903040])
 
gp: E = ellinit([0, -1, 0, 629536, 332903040])
 
magma: E := EllipticCurve([0, -1, 0, 629536, 332903040]);
 
oscar: E = EllipticCurve([0, -1, 0, 629536, 332903040])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(-334, 9234\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $4.0915810469557177063904867467$

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

Torsion generators

\( \left(-415, 0\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(-415, 0\right) \), \((-334,\pm 9234)\), \((13746,\pm 1614354)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 439824 \)  =  $2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-63904181593150660608 $  =  $-1 \cdot 2^{13} \cdot 3^{8} \cdot 7^{6} \cdot 11^{2} \cdot 17^{4} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{57258048889007}{132611470002} \)  =  $2^{-1} \cdot 3^{-8} \cdot 11^{-2} \cdot 17^{-4} \cdot 38543^{3}$
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: $2.4843616116201458270228486523\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.81825935653254386505294015912\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $4.0915810469557177063904867467\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.13670448285442641015760212710\dots$
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: $ 32 $  = $ 2\cdot2\cdot2\cdot2\cdot2 $
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: $2$
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'(E,1) $ ≈ $ 4.4746997686484317744767741238 $
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

$\displaystyle 4.474699769 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.136704 \cdot 4.091581 \cdot 32}{2^2} \approx 4.474699769$

# 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 439824.2.a.r

\( q - q^{3} - 2 q^{5} + q^{9} - q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 4 q^{19} + O(q^{20}) \) 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: 10616832
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{5}^{*}$ Additive -1 4 13 1
$3$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$7$ $2$ $I_0^{*}$ Additive -1 2 6 0
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $2$ $I_{4}$ Non-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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.16

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 = [[785, 280, 420, 169], [7, 6, 946, 947], [543, 0, 0, 951], [1, 0, 8, 1], [48, 791, 567, 384], [945, 8, 944, 9], [120, 875, 497, 526], [1, 8, 0, 1], [1, 4, 4, 17]]
 
GL(2,Integers(952)).subgroup(gens)
 
Gens := [[785, 280, 420, 169], [7, 6, 946, 947], [543, 0, 0, 951], [1, 0, 8, 1], [48, 791, 567, 384], [945, 8, 944, 9], [120, 875, 497, 526], [1, 8, 0, 1], [1, 4, 4, 17]];
 
sub<GL(2,Integers(952))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 952 = 2^{3} \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 785 & 280 \\ 420 & 169 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 946 & 947 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 951 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 48 & 791 \\ 567 & 384 \end{array}\right),\left(\begin{array}{rr} 945 & 8 \\ 944 & 9 \end{array}\right),\left(\begin{array}{rr} 120 & 875 \\ 497 & 526 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.

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

The torsion field $K:=\Q(E[952])$ is a degree-$5053612032$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/952\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 439824r consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 1122h4, its twist by $28$.

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.