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Properties

Label 462550f1
Conductor $462550$
Discriminant $-4.613\times 10^{24}$
j-invariant \( -\frac{982134513985}{19854792232} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-12731076x+104806732298\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-12731076xz^2+104806732298z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-16499473875x+4889912400528750\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 0, 1, -12731076, 104806732298])
 
Copy content gp:E = ellinit([1, 0, 1, -12731076, 104806732298])
 
Copy content magma:E := EllipticCurve([1, 0, 1, -12731076, 104806732298]);
 
Copy content oscar:E = elliptic_curve([1, 0, 1, -12731076, 104806732298])
 
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

\(\Z \oplus \Z\)

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$(-2598, 348211)$$2.2866268830820624341062630418$$\infty$
$(39452, 7791061)$$5.1366032496559499661308224643$$\infty$

Integral points

\( \left(-5348, 143886\right) \), \( \left(-5348, -138539\right) \), \( \left(-2598, 348211\right) \), \( \left(-2598, -345614\right) \), \( \left(39452, 7791061\right) \), \( \left(39452, -7830514\right) \) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 462550 \) = $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$
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: $\Delta$  =  $-4613317755157516590625000$ = $-1 \cdot 2^{3} \cdot 5^{8} \cdot 11^{2} \cdot 29^{11} $
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: $j$  =  \( -\frac{982134513985}{19854792232} \) = $-1 \cdot 2^{-3} \cdot 5 \cdot 11^{-2} \cdot 29^{-5} \cdot 5813^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $3.4135073334405569756867369547$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $0.65690081015791971236126138303$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $0.9330897987157233$
Szpiro ratio: $\sigma_{m}$ ≈ $4.927278508725074$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 2$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 2$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $11.049096509585135139539545342$
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: $\Omega$ ≈ $0.065013984391954677256466829908$
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: $\prod_{p}c_p$ = $ 12 $  = $ 1\cdot3\cdot2\cdot2 $
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^{(2)}(E,1)/2!$ ≈ $8.6201494562324265478782908520 $
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

$$\begin{aligned} 8.620149456 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.065014 \cdot 11.049097 \cdot 12}{1^2} \\ & \approx 8.620149456\end{aligned}$$

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, -12731076, 104806732298]); 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, -12731076, 104806732298]); 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 462550.2.a.f

\( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - q^{8} + q^{9} - q^{11} - 2 q^{12} - 2 q^{13} + q^{16} + 4 q^{17} - q^{18} + 6 q^{19} + O(q^{20}) \) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ 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: 90720000
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 $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$5$ $3$ $IV^{*}$ additive -1 2 8 0
$11$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$29$ $2$ $I_{5}^{*}$ additive 1 2 11 5

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 = [[89, 2, 89, 3], [1, 2, 0, 1], [1, 1, 231, 0], [1, 0, 2, 1], [117, 2, 117, 3], [231, 2, 230, 3], [175, 2, 175, 3]] GL(2,Integers(232)).subgroup(gens)
 
Copy content magma:Gens := [[89, 2, 89, 3], [1, 2, 0, 1], [1, 1, 231, 0], [1, 0, 2, 1], [117, 2, 117, 3], [231, 2, 230, 3], [175, 2, 175, 3]]; sub<GL(2,Integers(232))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 232 = 2^{3} \cdot 29 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 89 & 2 \\ 89 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 231 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 117 & 2 \\ 117 & 3 \end{array}\right),\left(\begin{array}{rr} 231 & 2 \\ 230 & 3 \end{array}\right),\left(\begin{array}{rr} 175 & 2 \\ 175 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[232])$ is a degree-$523837440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/232\Z)$.

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 $4$ \( 21025 = 5^{2} \cdot 29^{2} \)
$3$ good $2$ \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \)
$5$ additive $14$ \( 18502 = 2 \cdot 11 \cdot 29^{2} \)
$11$ nonsplit multiplicative $12$ \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \)
$29$ additive $450$ \( 550 = 2 \cdot 5^{2} \cdot 11 \)

Isogenies

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

This curve has no rational isogenies. Its isogeny class 462550f consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 15950e1, its twist by $145$.

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.