Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-30567022x-65049473804\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-30567022xz^2-65049473804z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-39614861187x-3034354026884994\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-3195, 4804\right)\) | \(\left(-3169, -673\right)\) |
$\hat{h}(P)$ | ≈ | $4.2785550290349802405899140854$ | $5.7021365448154724440446545391$ |
Torsion generators
\( \left(-\frac{12901}{4}, \frac{12901}{8}\right) \)
Integral points
\( \left(-3195, 4804\right) \), \( \left(-3195, -1609\right) \), \( \left(-3169, 3842\right) \), \( \left(-3169, -673\right) \), \( \left(7695, 390431\right) \), \( \left(7695, -398126\right) \), \( \left(17655, 2204479\right) \), \( \left(17655, -2222134\right) \)
Invariants
Conductor: | \( 431970 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $581319371106081263040 $ | = | $2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{6} \cdot 17^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1782900110862842086081}{328139630024640} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 527207^{3}$ | 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: | $2.9870329632092632393264050254\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.7880853268100779672954332364\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9994946520793547\dots$ | |||
Szpiro ratio: | $4.87972843732698\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $20.283741869315231555828221512\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.064208284524429321463031132137\dots$ | 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: | $ 32 $ = $ 2\cdot1\cdot1\cdot2\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! $ ≈ $ 10.419074153320578068149135068 $ | 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
$\displaystyle 10.419074153 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.064208 \cdot 20.283742 \cdot 32}{2^2} \approx 10.419074153$
Modular invariants
Modular form 431970.2.a.y
For more coefficients, see the Downloads section to the right.
Modular degree: | 47185920 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$17$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
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 | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 44880 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4093 & 20416 \\ 11484 & 31945 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11738 & 33671 \\ 37147 & 1926 \end{array}\right),\left(\begin{array}{rr} 36961 & 20416 \\ 14168 & 28689 \end{array}\right),\left(\begin{array}{rr} 10616 & 4081 \\ 44143 & 36730 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 44879 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 44782 & 44867 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 44876 & 44877 \end{array}\right),\left(\begin{array}{rr} 44865 & 16 \\ 44864 & 17 \end{array}\right),\left(\begin{array}{rr} 43528 & 4081 \\ 38159 & 36730 \end{array}\right)$.
The torsion field $K:=\Q(E[44880])$ is a degree-$3049493889024000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/44880\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 431970y
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3570t4, its twist by $-11$.
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.