Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+42047467x-11869259992\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+42047467xz^2-11869259992z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+54493517853x-553935674728674\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{1127}{4}, -\frac{1131}{8}\right) \)
Integral points
None
Invariants
| Conductor: | \( 400710 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-4818622652183916108533160 $ | = | $-1 \cdot 2^{3} \cdot 3^{6} \cdot 5 \cdot 19^{6} \cdot 37^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{174751791402194852399}{102423900876336360} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-1} \cdot 37^{-8} \cdot 5590799^{3}$ | 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: | $3.4256121125827495585097336637\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.9533926229995293285052199478\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0675015979740146\dots$ | |||
| Szpiro ratio: | $4.9822932949780325\dots$ | |||
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: | $0.045324451903492931604019327382\dots$ | 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: | $ 48 $ = $ 1\cdot( 2 \cdot 3 )\cdot1\cdot2^{2}\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)
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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: | $4$ = $2^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 2.1755736913676607169929277143 $ | 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
$\displaystyle 2.175573691 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.045324 \cdot 1.000000 \cdot 48}{2^2} \approx 2.175573691$
Modular invariants
Modular form 400710.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 95551488 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $37$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168720 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 168716 & 168717 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 168705 & 16 \\ 168704 & 17 \end{array}\right),\left(\begin{array}{rr} 121373 & 124336 \\ 91504 & 159525 \end{array}\right),\left(\begin{array}{rr} 58616 & 71041 \\ 78223 & 133210 \end{array}\right),\left(\begin{array}{rr} 95476 & 113221 \\ 75639 & 48850 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 168622 & 168707 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 54398 & 113221 \\ 149929 & 48850 \end{array}\right),\left(\begin{array}{rr} 71039 & 0 \\ 0 & 168719 \end{array}\right),\left(\begin{array}{rr} 150481 & 124336 \\ 84968 & 151089 \end{array}\right)$.
The torsion field $K:=\Q(E[168720])$ is a degree-$661624187151974400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 400710.x
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1110.k6, its twist by $-19$.
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 $0$.