Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-70281x-6250343\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-70281xz^2-6250343z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1124499x-401146450\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-148, -871\right)\) | \(\left(-123, 784\right)\) |
$\hat{h}(P)$ | ≈ | $2.3322862225806024336074010674$ | $2.4381515526013136295948792558$ |
Torsion generators
\( \left(-106, 53\right) \), \( \left(302, -151\right) \)
Integral points
\( \left(-148, 1019\right) \), \( \left(-148, -871\right) \), \( \left(-123, 784\right) \), \( \left(-123, -661\right) \), \( \left(-114, 577\right) \), \( \left(-114, -463\right) \), \( \left(-106, 53\right) \), \( \left(302, -151\right) \), \( \left(419, 5933\right) \), \( \left(419, -6352\right) \), \( \left(744, 18413\right) \), \( \left(744, -19157\right) \), \( \left(761, 19127\right) \), \( \left(761, -19888\right) \), \( \left(2036, 90017\right) \), \( \left(2036, -92053\right) \), \( \left(5072252, 11421016859\right) \), \( \left(5072252, -11426089111\right) \)
Invariants
Conductor: | \( 473382 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $5245734934082916 $ | = | $2^{2} \cdot 3^{8} \cdot 7^{2} \cdot 13^{2} \cdot 17^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2181825073}{298116} \) | = | $2^{-2} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 1297^{3}$ | 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: | $1.7433118802327227305121657426\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.22260093612944015531022418480\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9566415663757943\dots$ | |||
Szpiro ratio: | $3.4508388200182707\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.3623425120203151243850566371\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.29587320270462964624323402110\dots$ | 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: | $ 128 $ = $ 2\cdot2^{2}\cdot2\cdot2\cdot2^{2} $ | 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: | $4$ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 12.692587624245117057446116086 $ | 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
$\displaystyle 12.692587624 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.295873 \cdot 5.362343 \cdot 128}{4^2} \approx 12.692587624$
Modular invariants
Modular form 473382.2.a.bs
For more coefficients, see the Downloads section to the right.
Modular degree: | 3538944 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$13$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$17$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 37128 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 29955 & 8738 \\ 1870 & 28391 \end{array}\right),\left(\begin{array}{rr} 18565 & 8738 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 27847 & 17476 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37125 & 4 \\ 37124 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 24751 & 19652 \\ 3638 & 2175 \end{array}\right),\left(\begin{array}{rr} 8735 & 0 \\ 0 & 37127 \end{array}\right),\left(\begin{array}{rr} 8569 & 17476 \\ 25874 & 34953 \end{array}\right)$.
The torsion field $K:=\Q(E[37128])$ is a degree-$6357363078463488$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/37128\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 473382bs
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 546g2, its twist by $-51$.
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.