Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-17768255440x-911677835353480\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-17768255440xz^2-911677835353480z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-23027659050267x-42535172003274812106\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{615679}{4}, -\frac{615679}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 447330 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 31 \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-41986862587247533428179923320 $ | = | $-1 \cdot 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31^{16} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{620380415613717203478802928936674561}{41986862587247533428179923320} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-1} \cdot 31^{-16} \cdot 37^{-1} \cdot 27767^{3} \cdot 30715463^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $4.5470967845776061272217776877\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $4.5470967845776061272217776877\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0088742755855573\dots$ | |||
Szpiro ratio: | $6.334288632897817\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.0065381758668661594228483407332\dots$ | 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);
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Tamagawa product: | $ 6 $ = $ 3\cdot1\cdot1\cdot1\cdot2\cdot1 $ | 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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1024$ = $32^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 10.042638131506420873495051366 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 10.042638132 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1024 \cdot 0.006538 \cdot 1.000000 \cdot 6}{2^2} \approx 10.042638132$
Modular invariants
Modular form 447330.2.a.cv
For more coefficients, see the Downloads section to the right.
Modular degree: | 880803840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$13$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$31$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
$37$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 16.48.0.44 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7157280 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \cdot 37 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 7154718 & 7155275 \end{array}\right),\left(\begin{array}{rr} 5505622 & 3 \\ 3852365 & 7157068 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 2862944 & 29 \\ 2865739 & 2562 \end{array}\right),\left(\begin{array}{rr} 773776 & 25 \\ 195479 & 3186 \end{array}\right),\left(\begin{array}{rr} 4473314 & 31 \\ 4472189 & 7154820 \end{array}\right),\left(\begin{array}{rr} 4771552 & 29 \\ 2388587 & 2562 \end{array}\right),\left(\begin{array}{rr} 5310241 & 32 \\ 6233776 & 513 \end{array}\right),\left(\begin{array}{rr} 6709951 & 32 \\ 1341990 & 1 \end{array}\right),\left(\begin{array}{rr} 7157249 & 32 \\ 7157248 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[7157280])$ is a degree-$502956956821751857152000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7157280\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 447330cv
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
This elliptic curve is its own minimal quadratic twist.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.