Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+113542800x+297499292125\)
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(homogenize, simplify) |
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\(y^2z=x^3+113542800xz^2+297499292125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+113542800x+297499292125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-1945, 263250\right)\)
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| $\hat{h}(P)$ | ≈ | $4.3478802688982825320695631187$ |
Torsion generators
\( \left(-2485, 0\right) \)
Integral points
\( \left(-2485, 0\right) \), \((-1945,\pm 263250)\)
Invariants
| Conductor: | \( 485100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-131917083150101525718750000 $ | = | $-1 \cdot 2^{4} \cdot 3^{10} \cdot 5^{9} \cdot 7^{9} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{7549996227362816}{6152409907875} \) | = | $2^{20} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-6} \cdot 1931^{3}$ | 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: | $3.6998273143448474928122000689\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.1417980790794373707891107050\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0540916531253661\dots$ | |||
| Szpiro ratio: | $5.137191221632644\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $4.3478802688982825320695631187\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.037743834849732509186483968602\dots$ | 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: | $ 192 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot2\cdot( 2 \cdot 3 ) $ | 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'(E,1) $ ≈ $ 7.8770723911539527668260247738 $ | 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
$\displaystyle 7.877072391 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.037744 \cdot 4.347880 \cdot 192}{2^2} \approx 7.877072391$
Modular invariants
Modular form 485100.2.a.he
For more coefficients, see the Downloads section to the right.
| Modular degree: | 143327232 | 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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
| $7$ | $2$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
| $11$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 4609 & 12 \\ 4608 & 13 \end{array}\right),\left(\begin{array}{rr} 2630 & 4617 \\ 3987 & 8 \end{array}\right),\left(\begin{array}{rr} 3079 & 4608 \\ 3080 & 4619 \end{array}\right),\left(\begin{array}{rr} 4239 & 2698 \\ 1946 & 401 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 4570 & 4611 \end{array}\right),\left(\begin{array}{rr} 4610 & 4617 \\ 2799 & 8 \end{array}\right),\left(\begin{array}{rr} 2521 & 12 \\ 1266 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[4620])$ is a degree-$613122048000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4620\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 485100.he
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4620.j4, its twist by $105$.
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.