↑

Properties

Label 446160.ft2
Conductor $446160$
Discriminant $9.154\times 10^{21}$
j-invariant \( \frac{182864522286982801}{463015182960} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-31974856x-69450519436\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-31974856xz^2-69450519436z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2589963363x-50621658778782\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 0, -31974856, -69450519436])
 
gp: E = ellinit([0, 1, 0, -31974856, -69450519436])
 
magma: E := EllipticCurve([0, 1, 0, -31974856, -69450519436]);
 
oscar: E = EllipticCurve([0, 1, 0, -31974856, -69450519436])
 
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

$P$ =  \(\left(-3268, 12342\right)\) Copy content Toggle raw display \(\left(-\frac{78796}{25}, \frac{665742}{125}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.3321930067785624914661877062$$5.5451123327236267188499608682$

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

Torsion generators

\( \left(-3389, 0\right) \) 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

\( \left(-3389, 0\right) \), \((-3268,\pm 12342)\), \((10067,\pm 793092)\), \((17060,\pm 2085798)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 446160 \)  =  $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $9154092450807704125440 $  =  $2^{16} \cdot 3^{3} \cdot 5 \cdot 11^{8} \cdot 13^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{182864522286982801}{463015182960} \)  =  $2^{-4} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-8} \cdot 567601^{3}$
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: $3.0908078925836712735339162544\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $1.1151860332929575960899404122\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.0750149448569075\dots$
Szpiro ratio: $4.877988355978908\dots$

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $12.735091662300222142624690896\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.063498503262424767650337806626\dots$
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: $ 96 $  = $ 2\cdot3\cdot1\cdot2^{3}\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^{(2)}(E,1)/2! $ ≈ $ 19.407822227180378714616250113 $
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

$\displaystyle 19.407822227 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.063499 \cdot 12.735092 \cdot 96}{2^2} \approx 19.407822227$

# 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 446160.2.a.ft

\( q + q^{3} - q^{5} + q^{9} + q^{11} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20}) \) 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: 35389440
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
comment: Manin constant
 
magma: ManinConstant(E);
 
* The Manin constant is correct provided that curve 446160.ft6 is optimal.

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$(j)_{-}$
$2$ $2$ $I_{8}^{*}$ Additive -1 4 16 4
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$13$ $2$ $I_0^{*}$ Additive 1 2 6 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
$2$ 2B 8.12.0.5

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 = [[31679, 0, 0, 34319], [24961, 26416, 6968, 5409], [31342, 27053, 18785, 3938], [1, 16, 0, 1], [5, 4, 34316, 34317], [15, 2, 34222, 34307], [6176, 21125, 15795, 5266], [29576, 31681, 4303, 10570], [1, 0, 16, 1], [1, 26416, 21788, 13989], [34305, 16, 34304, 17]]
 
GL(2,Integers(34320)).subgroup(gens)
 
Gens := [[31679, 0, 0, 34319], [24961, 26416, 6968, 5409], [31342, 27053, 18785, 3938], [1, 16, 0, 1], [5, 4, 34316, 34317], [15, 2, 34222, 34307], [6176, 21125, 15795, 5266], [29576, 31681, 4303, 10570], [1, 0, 16, 1], [1, 26416, 21788, 13989], [34305, 16, 34304, 17]];
 
sub<GL(2,Integers(34320))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 34320 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 31679 & 0 \\ 0 & 34319 \end{array}\right),\left(\begin{array}{rr} 24961 & 26416 \\ 6968 & 5409 \end{array}\right),\left(\begin{array}{rr} 31342 & 27053 \\ 18785 & 3938 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 34316 & 34317 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 34222 & 34307 \end{array}\right),\left(\begin{array}{rr} 6176 & 21125 \\ 15795 & 5266 \end{array}\right),\left(\begin{array}{rr} 29576 & 31681 \\ 4303 & 10570 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 26416 \\ 21788 & 13989 \end{array}\right),\left(\begin{array}{rr} 34305 & 16 \\ 34304 & 17 \end{array}\right)$.

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

The torsion field $K:=\Q(E[34320])$ is a degree-$1020235087872000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/34320\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 446160.ft consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 330.d2, its twist by $-52$.

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.