Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1740531x+218210434\)
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(homogenize, simplify) |
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\(y^2z=x^3-1740531xz^2+218210434z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1740531x+218210434\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(-325, 27378\right)\)
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\(\left(-1339, 12168\right)\)
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| $\hat{h}(P)$ | ≈ | $1.2323218684163029617387309381$ | $2.2426241144005333658847326219$ |
Torsion generators
\( \left(-1378, 0\right) \)
Integral points
\( \left(-1378, 0\right) \), \((-1339,\pm 12168)\), \((-511,\pm 31212)\), \((-325,\pm 27378)\), \((1326,\pm 15548)\), \((1889,\pm 60588)\), \((2379,\pm 97682)\), \((3185,\pm 164268)\), \((100061,\pm 31648968)\)
Invariants
| Conductor: | \( 413712 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $316892277287518712832 $ | = | $2^{10} \cdot 3^{10} \cdot 13^{7} \cdot 17^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{161838334948}{87947613} \) | = | $2^{2} \cdot 3^{-4} \cdot 13^{-1} \cdot 17^{-4} \cdot 3433^{3}$ | 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: | $2.6246547448032351577154384020\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.21525127127179085281004529488\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9341909408008048\dots$ | |||
| Szpiro ratio: | $4.231270490374442\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $2.4815240818366197013112512996\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.14981738761711957345774704273\dots$ | 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^{2}\cdot2^{2}\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 11.896814567991475439082056293 $ | 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
$\displaystyle 11.896814568 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.149817 \cdot 2.481524 \cdot 128}{2^2} \approx 11.896814568$
Modular invariants
Modular form 413712.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16515072 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | Additive | 1 | 4 | 10 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
| $17$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
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 \( 5304 = 2^{3} \cdot 3 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 667 & 666 \\ 2002 & 4651 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 5298 & 5299 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5297 & 8 \\ 5296 & 9 \end{array}\right),\left(\begin{array}{rr} 4637 & 4640 \\ 1966 & 4635 \end{array}\right),\left(\begin{array}{rr} 400 & 5301 \\ 4891 & 5302 \end{array}\right),\left(\begin{array}{rr} 1873 & 8 \\ 2188 & 33 \end{array}\right),\left(\begin{array}{rr} 3535 & 5296 \\ 3532 & 5271 \end{array}\right)$.
The torsion field $K:=\Q(E[5304])$ is a degree-$3153453907968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5304\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 413712.v
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 5304.b2, its twist by $156$.
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.