Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-2438100x+1465970886\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-2438100xz^2+1465970886z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-39009603x+93783127102\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(945, 1809\right)\)
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\(\left(\frac{2727}{4}, \frac{85023}{8}\right)\)
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| $\hat{h}(P)$ | ≈ | $0.58962876551243258393311306158$ | $1.2760511487994472849310734121$ |
Integral points
\( \left(-537, 51456\right) \), \( \left(-537, -50919\right) \), \( \left(867, 1380\right) \), \( \left(867, -2247\right) \), \( \left(945, 1809\right) \), \( \left(945, -2754\right) \), \( \left(2457, 100278\right) \), \( \left(2457, -102735\right) \), \( \left(3213, 162081\right) \), \( \left(3213, -165294\right) \)
Invariants
| Conductor: | \( 422370 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-82445181342468750 $ | = | $-1 \cdot 2 \cdot 3^{9} \cdot 5^{6} \cdot 13^{5} \cdot 19^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{6090369270477666841}{313278468750} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-3} \cdot 5^{-6} \cdot 13^{-5} \cdot 19 \cdot 684379^{3}$ | 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: | $2.3153663645884097512380661343\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.2753203903932814955389389439\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9836222285434395\dots$ | |||
| Szpiro ratio: | $4.302566222973559\dots$ | |||
BSD invariants
| Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.75238036366090414579591347289\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.32271785367875182875294261031\dots$ | 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\cdot2^{2}\cdot2\cdot5\cdot1 $ | 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: | $1$ | 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! $ ≈ $ 9.7122630444274301489079547279 $ | 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
$\displaystyle 9.712263044 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.322718 \cdot 0.752380 \cdot 40}{1^2} \approx 9.712263044$
Modular invariants
Modular form 422370.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10056960 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
| $19$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 145 & 2 \\ 145 & 3 \end{array}\right),\left(\begin{array}{rr} 311 & 2 \\ 310 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 311 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 209 & 2 \\ 209 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$966131712$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/312\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 422370.s consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 140790.cv1, its twist by $-3$.
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.