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Properties

Label 446490z2
Conductor $446490$
Discriminant $-3.231\times 10^{19}$
j-invariant \( -\frac{1642140750896013526054039801}{25018323000} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-26766408510x-1685510810183700\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-26766408510xz^2-1685510810183700z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-428262536163x-107873120114292962\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, -1, 0, -26766408510, -1685510810183700])
 
Copy content gp:E = ellinit([1, -1, 0, -26766408510, -1685510810183700])
 
Copy content magma:E := EllipticCurve([1, -1, 0, -26766408510, -1685510810183700]);
 
Copy content oscar:E = elliptic_curve([1, -1, 0, -26766408510, -1685510810183700])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(\frac{1292312230744336940941988728182572695932043277781}{6810395745928361089103895255153174727388176}, \frac{144840233850337438943725143339833092875896574532684298016638149457579145}{17772909262637559451512754596691297429334040255033996228782192576}\right) \)$107.41334143774607969913429744$$\infty$

$P$$\hat{h}(P)$Order
\([3372512974704515241135888054244615016009617129618368621919545909728556:144840233850337438943725143339833092875896574532684298016638149457579145:17772909262637559451512754596691297429334040255033996228782192576]\)$107.41334143774607969913429744$$\infty$

$P$$\hat{h}(P)$Order
\( \left(\frac{1292310528145400458851716452208758907638361430737}{1702598936482090272275973813788293681847044}, \frac{146526490337689696564293087366955400383901383097493482327597922412443423}{2221613657829694931439094324586412178666755031879249528597774072}\right) \)$107.41334143774607969913429744$$\infty$

Integral points

None

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

Invariants

Conductor: $N$  =  \( 446490 \) = $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 41$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Minimal Discriminant: $\Delta$  =  $-32310362792595987000$ = $-1 \cdot 2^{3} \cdot 3^{7} \cdot 5^{3} \cdot 11^{8} \cdot 41^{3} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -\frac{1642140750896013526054039801}{25018323000} \) = $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-2} \cdot 41^{-3} \cdot 107^{3} \cdot 1187^{3} \cdot 1327^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $4.2232492135814122381807537993$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $2.4749954328481721204521593919$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0150910765771144$
Szpiro ratio: $\sigma_{m}$ ≈ $6.4296822110543745$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $107.41334143774607969913429744$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $0.0059016307530551777113367814819$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 8 $  = $ 1\cdot2^{2}\cdot1\cdot2\cdot1 $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $1$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L'(E,1)$ ≈ $5.0713110329393465704413684740 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  ≈  $1$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 5.071311033 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.005902 \cdot 107.413341 \cdot 8}{1^2} \\ & \approx 5.071311033\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, -1, 0, -26766408510, -1685510810183700]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, -1, 0, -26766408510, -1685510810183700]); 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 446490.2.a.z

\( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + 4 q^{13} - q^{14} + q^{16} - 5 q^{19} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 415549440
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

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

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$3$ $4$ $I_{1}^{*}$ additive -1 2 7 1
$5$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$11$ $2$ $I_{2}^{*}$ additive -1 2 8 2
$41$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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 $\ell$-adic index
$3$ 3B 3.4.0.1 $4$

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[4, 3, 9, 7], [37102, 41613, 19679, 20294], [43297, 44286, 16731, 24619], [34439, 0, 0, 54119], [27061, 44286, 22143, 24619], [40591, 44286, 0, 1], [54115, 6, 54114, 7], [1, 6, 0, 1], [3, 4, 8, 11], [21121, 44286, 4323, 24619], [1, 0, 6, 1]] GL(2,Integers(54120)).subgroup(gens)
 
Copy content magma:Gens := [[4, 3, 9, 7], [37102, 41613, 19679, 20294], [43297, 44286, 16731, 24619], [34439, 0, 0, 54119], [27061, 44286, 22143, 24619], [40591, 44286, 0, 1], [54115, 6, 54114, 7], [1, 6, 0, 1], [3, 4, 8, 11], [21121, 44286, 4323, 24619], [1, 0, 6, 1]]; sub<GL(2,Integers(54120))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 54120 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 41 \), index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 37102 & 41613 \\ 19679 & 20294 \end{array}\right),\left(\begin{array}{rr} 43297 & 44286 \\ 16731 & 24619 \end{array}\right),\left(\begin{array}{rr} 34439 & 0 \\ 0 & 54119 \end{array}\right),\left(\begin{array}{rr} 27061 & 44286 \\ 22143 & 24619 \end{array}\right),\left(\begin{array}{rr} 40591 & 44286 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 54115 & 6 \\ 54114 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 21121 & 44286 \\ 4323 & 24619 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[54120])$ is a degree-$80441612697600000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/54120\Z)$.

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
$2$ nonsplit multiplicative $4$ \( 223245 = 3^{2} \cdot 5 \cdot 11^{2} \cdot 41 \)
$3$ additive $8$ \( 121 = 11^{2} \)
$5$ nonsplit multiplicative $6$ \( 89298 = 2 \cdot 3^{2} \cdot 11^{2} \cdot 41 \)
$11$ additive $72$ \( 3690 = 2 \cdot 3^{2} \cdot 5 \cdot 41 \)
$41$ nonsplit multiplicative $42$ \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 446490z consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 13530j2, 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.