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Properties

Label 477950z1
Conductor $477950$
Discriminant $-1.639\times 10^{23}$
j-invariant \( -\frac{2282033126907134425}{647168} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-17354350951x-879956879831702\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-17354350951xz^2-879956879831702z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-22491238831875x-41055200711711381250\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -17354350951, -879956879831702])
 
Copy content gp:E = ellinit([1, 0, 1, -17354350951, -879956879831702])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -17354350951, -879956879831702]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -17354350951, -879956879831702])
 
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$$\hat{h}(P)$Order
$(56574703694916774279864628583132728233294865864481189751219850357324243169288485020774193411170561726251047037727125597227921796466972517864679704002266438255328532761204619674325014097507359389862070928854107852837651447006284836175969417142057916054272257653795302564861638779470944589301720273339126/50887848691402861632702170510019612563136705008618847975650422107712765982494708425661114510929133611448502058912392184266869407644734653499968671528019861747424754819834917026740732971956273563532366573205675076782098839474260513022402290676340327719818732111218325625383154808677144028168890625, 422196226362037065235622466898871147113334475201410966858543800280761369460941862384511959326099319560254603476814128347478276955577000714700198711526158802941866347839137693515591184210739924211547544975435318882327925702770800108913227015016013073547172504064792334666757701379834992537093781567952382520894454465124904224737700032598214576599042194972477305004171142640854407625835794791122881757396732908005226133684543037760471080236360739043130376/363012129755381391694240130389478092138974031523956921812704061516370768973622882874747904667741098741722715725080780395965395860037541665968154880827785502487662897236622909261744697588721842411690763548080870520261909272972556345570394041305682052652933300889504093976345396927105385570283482644378392149227496225835147436898298106986924126506500341910267738065568219324474657388440101735675327989160863335748351560409571736437314634037109375)$$694.87726381005310739426387471$$\infty$

Integral points

None

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

Invariants

Conductor: $N$  =  \( 477950 \) = $2 \cdot 5^{2} \cdot 11^{2} \cdot 79$
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)
 
Discriminant: $\Delta$  =  $-163924523478320000000000$ = $-1 \cdot 2^{13} \cdot 5^{10} \cdot 11^{10} \cdot 79 $
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: $j$  =  \( -\frac{2282033126907134425}{647168} \) = $-1 \cdot 2^{-13} \cdot 5^{2} \cdot 11^{2} \cdot 79^{-1} \cdot 91033^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $4.2613698215598709310891988597$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $0.92192550053281183220361310037$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9894268827023249$
Szpiro ratio: $\sigma_{m}$ ≈ $6.296802346756302$

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: $\mathrm{Reg}(E/\Q)$ ≈ $694.87726381005310739426387471$
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: $\Omega$ ≈ $0.0065768428546017148721814720439$
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: $\prod_{p}c_p$ = $ 1 $
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)$ ≈ $4.5700985673143385767017360601 $
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

$$\begin{aligned} 4.570098567 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.006577 \cdot 694.877264 \cdot 1}{1^2} \\ & \approx 4.570098567\end{aligned}$$

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, 0, 1, -17354350951, -879956879831702]); 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, 0, 1, -17354350951, -879956879831702]); 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 477950.2.a.z

\( q - q^{2} + q^{3} + q^{4} - q^{6} - 4 q^{7} - q^{8} - 2 q^{9} + q^{12} - 4 q^{13} + 4 q^{14} + q^{16} + 4 q^{17} + 2 q^{18} + O(q^{20}) \) Copy content Toggle raw display

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

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

Modular degree: 543628800
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: yes
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 4 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{13}$ nonsplit multiplicative 1 1 13 13
$5$ $1$ $II^{*}$ additive 1 2 10 0
$11$ $1$ $II^{*}$ additive -1 2 10 0
$79$ $1$ $I_{1}$ split multiplicative -1 1 1 1

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 $\ell$.

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 = [[1, 1, 631, 0], [161, 2, 161, 3], [631, 2, 630, 3], [317, 2, 317, 3], [1, 0, 2, 1], [1, 2, 0, 1], [159, 2, 0, 1]] GL(2,Integers(632)).subgroup(gens)
 
Copy content magma:Gens := [[1, 1, 631, 0], [161, 2, 161, 3], [631, 2, 630, 3], [317, 2, 317, 3], [1, 0, 2, 1], [1, 2, 0, 1], [159, 2, 0, 1]]; sub<GL(2,Integers(632))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 632 = 2^{3} \cdot 79 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 1 \\ 631 & 0 \end{array}\right),\left(\begin{array}{rr} 161 & 2 \\ 161 & 3 \end{array}\right),\left(\begin{array}{rr} 631 & 2 \\ 630 & 3 \end{array}\right),\left(\begin{array}{rr} 317 & 2 \\ 317 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 159 & 2 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[632])$ is a degree-$29530275840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/632\Z)$.

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 $4$ \( 238975 = 5^{2} \cdot 11^{2} \cdot 79 \)
$5$ additive $2$ \( 19118 = 2 \cdot 11^{2} \cdot 79 \)
$11$ additive $32$ \( 3950 = 2 \cdot 5^{2} \cdot 79 \)
$13$ good $2$ \( 238975 = 5^{2} \cdot 11^{2} \cdot 79 \)
$79$ split multiplicative $80$ \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)

Isogenies

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

This curve has no rational isogenies. Its isogeny class 477950z consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 477950l1, its twist by $-55$.

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.