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Properties

Label 413820.be1
Conductor $413820$
Discriminant $8.415\times 10^{18}$
j-invariant \( \frac{5405726654464}{407253125} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-1003332x+360768881\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-1003332xz^2+360768881z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1003332x+360768881\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, -1003332, 360768881])
 
gp: E = ellinit([0, 0, 0, -1003332, 360768881])
 
magma: E := EllipticCurve([0, 0, 0, -1003332, 360768881]);
 
oscar: E = EllipticCurve([0, 0, 0, -1003332, 360768881])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(242, 11495\right)\) Copy content Toggle raw display \(\left(-1088, 12825\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.34104732803646438817367681816$$1.7278249876279703532643017017$

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

Torsion generators

\( \left(451, 0\right) \) 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

\((-1088,\pm 12825)\), \((-638,\pm 27225)\), \((-88,\pm 21175)\), \((242,\pm 11495)\), \((280,\pm 10089)\), \( \left(451, 0\right) \), \((812,\pm 9025)\), \((847,\pm 10890)\), \((1292,\pm 34945)\), \((2332,\pm 103455)\), \((2750,\pm 135641)\), \((8470,\pm 774279)\), \((9647,\pm 942590)\), \((322102,\pm 182804985)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 413820 \)  =  $2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 19$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $8415269859402450000 $  =  $2^{4} \cdot 3^{6} \cdot 5^{5} \cdot 11^{6} \cdot 19^{4} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{5405726654464}{407253125} \)  =  $2^{14} \cdot 5^{-5} \cdot 19^{-4} \cdot 691^{3}$
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: $2.3766272279326684310723504511\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.39732438701277987687134533650\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.9907756543685527\dots$
Szpiro ratio: $4.103405962670183\dots$

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.58669052209510745426907576432\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.22758589706308422597481927734\dots$
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: $ 480 $  = $ 3\cdot2\cdot5\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: $2$
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^{(2)}(E,1)/2! $ ≈ $ 16.022698652330912011721585760 $
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

$\displaystyle 16.022698652 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.227586 \cdot 0.586691 \cdot 480}{2^2} \approx 16.022698652$

# 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 413820.2.a.be

\( q + q^{5} - 2 q^{7} - 6 q^{13} + 2 q^{17} + q^{19} + O(q^{20}) \) 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: 8294400
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 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $3$ $IV$ Additive -1 2 4 0
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$5$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$11$ $4$ $I_0^{*}$ Additive -1 2 6 0
$19$ $4$ $I_{4}$ 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.3

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 = [[22441, 19008, 11484, 793], [1, 8, 0, 1], [12541, 19008, 0, 1], [9736, 22803, 21285, 24322], [1, 0, 8, 1], [5, 8, 48, 77], [25073, 8, 25072, 9], [8359, 0, 0, 25079], [15959, 0, 0, 25079], [1, 4, 4, 17], [14257, 20526, 4554, 4555], [3, 8, 28, 75]]
 
GL(2,Integers(25080)).subgroup(gens)
 
Gens := [[22441, 19008, 11484, 793], [1, 8, 0, 1], [12541, 19008, 0, 1], [9736, 22803, 21285, 24322], [1, 0, 8, 1], [5, 8, 48, 77], [25073, 8, 25072, 9], [8359, 0, 0, 25079], [15959, 0, 0, 25079], [1, 4, 4, 17], [14257, 20526, 4554, 4555], [3, 8, 28, 75]];
 
sub<GL(2,Integers(25080))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25080 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 22441 & 19008 \\ 11484 & 793 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12541 & 19008 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9736 & 22803 \\ 21285 & 24322 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 25073 & 8 \\ 25072 & 9 \end{array}\right),\left(\begin{array}{rr} 8359 & 0 \\ 0 & 25079 \end{array}\right),\left(\begin{array}{rr} 15959 & 0 \\ 0 & 25079 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 14257 & 20526 \\ 4554 & 4555 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right)$.

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

The torsion field $K:=\Q(E[25080])$ is a degree-$1198215659520000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25080\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 413820.be consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 380.b1, its twist by $33$.

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.