Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-9602155x-11453315932\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-9602155xz^2-11453315932z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-12444392259x-534328574934978\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{7161}{4}, \frac{7157}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $65632013677435338 $ | = | $2 \cdot 3^{5} \cdot 7 \cdot 19^{4} \cdot 23^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{661397832743623417}{443352042} \) | = | $2^{-1} \cdot 3^{-5} \cdot 7^{-1} \cdot 13^{3} \cdot 19^{-4} \cdot 67021^{3}$ | 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: | $2.5430488703318369562925358799\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.97530176236726211088915946400\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0038346664696574\dots$ | |||
Szpiro ratio: | $4.620215310581483\dots$ |
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: | $0.085764468297625600840941477303\dots$ | 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: | $ 40 $ = $ 1\cdot5\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) $ ≈ $ 3.4305787319050240336376590921 $ | 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
$\displaystyle 3.430578732 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.085764 \cdot 1.000000 \cdot 40}{2^2} \approx 3.430578732$
Modular invariants
Modular form 422142.2.a.bw
For more coefficients, see the Downloads section to the right.
Modular degree: | 15769600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$19$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$23$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 73416 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 73409 & 8 \\ 73408 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 31919 & 0 \\ 0 & 73415 \end{array}\right),\left(\begin{array}{rr} 61848 & 73025 \\ 37099 & 72198 \end{array}\right),\left(\begin{array}{rr} 68104 & 60651 \\ 52141 & 44690 \end{array}\right),\left(\begin{array}{rr} 42505 & 63848 \\ 55108 & 35145 \end{array}\right),\left(\begin{array}{rr} 46691 & 72220 \\ 72266 & 67437 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 73410 & 73411 \end{array}\right),\left(\begin{array}{rr} 29188 & 44689 \\ 20999 & 3198 \end{array}\right)$.
The torsion field $K:=\Q(E[73416])$ is a degree-$101857916784476160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/73416\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 422142.bw
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798.c1, its twist by $-23$.
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 $0$.