Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-23222948x+42991941831\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-23222948xz^2+42991941831z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-371567163x+2751112710038\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(3185, 34857\right)\) |
$\hat{h}(P)$ | ≈ | $2.0059262063557112291379456852$ |
Torsion generators
\( \left(-5563, 2781\right) \), \( \left(2885, -1443\right) \)
Integral points
\( \left(-5563, 2781\right) \), \( \left(-679, 242097\right) \), \( \left(-679, -241419\right) \), \( \left(2885, -1443\right) \), \( \left(3185, 34857\right) \), \( \left(3185, -38043\right) \), \( \left(8693, 701325\right) \), \( \left(8693, -710019\right) \)
Invariants
Conductor: | \( 402930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $3299548990796720640000 $ | = | $2^{12} \cdot 3^{12} \cdot 5^{4} \cdot 11^{6} \cdot 37^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1072487167529950801}{2554882560000} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-4} \cdot 37^{-2} \cdot 1023601^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.0084992231728241368165822337\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.2602454424395840190879878263\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0707325514302872\dots$ | |||
Szpiro ratio: | $4.84217019856706\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.0059262063557112291379456852\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14171909857162671094863049766\dots$ | 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: | $ 768 $ = $ ( 2^{2} \cdot 3 )\cdot2^{2}\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: | $4$ | 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) $ ≈ $ 13.645346580764844589760642645 $ | 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
$\displaystyle 13.645346581 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.141719 \cdot 2.005926 \cdot 768}{4^2} \approx 13.645346581$
Modular invariants
Modular form 402930.2.a.dj
For more coefficients, see the Downloads section to the right.
Modular degree: | 35389440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$5$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$11$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$37$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2Cs | 4.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 34033 & 22198 \\ 44418 & 31085 \end{array}\right),\left(\begin{array}{rr} 19537 & 4444 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48833 & 8 \\ 48832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 28865 & 43296 \\ 7744 & 37709 \end{array}\right),\left(\begin{array}{rr} 4445 & 18876 \\ 44374 & 25499 \end{array}\right),\left(\begin{array}{rr} 13319 & 0 \\ 0 & 48839 \end{array}\right),\left(\begin{array}{rr} 9967 & 31086 \\ 42834 & 17755 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 48836 & 48837 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$4433397940224000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 402930.dj
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1110.k4, its twist by $33$.
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.