Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-33075x-1157625\)
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(homogenize, simplify) |
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\(y^2z=x^3-33075xz^2-1157625z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-33075x-1157625\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-129, 981\right)\)
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| $\hat{h}(P)$ | ≈ | $3.4923989945475995553993445626$ |
Integral points
\((-129,\pm 981)\)
Invariants
| Conductor: | \( 793800 \) | = | $2^{3} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $1736763950250000 $ | = | $2^{4} \cdot 3^{10} \cdot 5^{6} \cdot 7^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( 2304 \) | = | $2^{8} \cdot 3^{2}$ | 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: | $1.6203921320531989813843046143\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.3038411994349143711038664953\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.7737056144690831\dots$ | |||
| Szpiro ratio: | $3.1530735950482316\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $3.4923989945475995553993445626\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.36829981603298818371755650368\dots$ | 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: | $ 6 $ = $ 2\cdot3\cdot1\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'(E,1) $ ≈ $ 7.7174994432340429148544374234 $ | 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
$\displaystyle 7.717499443 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.368300 \cdot 3.492399 \cdot 6}{1^2} \approx 7.717499443$
Modular invariants
Modular form 793800.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3326400 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(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$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | Additive | 1 | 3 | 4 | 0 |
| $3$ | $3$ | $IV^{*}$ | Additive | -1 | 4 | 10 | 0 |
| $5$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2Cn | 2.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 489 & 35 \\ 245 & 1224 \end{array}\right),\left(\begin{array}{rr} 179 & 0 \\ 0 & 1259 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 1120 \\ 350 & 911 \end{array}\right),\left(\begin{array}{rr} 1257 & 4 \\ 1256 & 5 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 1259 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 1252 & 1255 \end{array}\right)$.
The torsion field $K:=\Q(E[1260])$ is a degree-$30098718720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1260\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 793800.f consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 648.a1, 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.