Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-3225x+698625\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-3225xz^2+698625z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4180275x+32657748750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-15, 870\right)\) | \(\left(-60, 855\right)\) |
$\hat{h}(P)$ | ≈ | $0.45224518987196395300320546435$ | $1.3287606322034705289242325408$ |
Torsion generators
\( \left(-\frac{405}{4}, \frac{405}{8}\right) \)
Integral points
\( \left(-84, 663\right) \), \( \left(-84, -579\right) \), \( \left(-60, 855\right) \), \( \left(-60, -795\right) \), \( \left(-40, 895\right) \), \( \left(-40, -855\right) \), \( \left(-15, 870\right) \), \( \left(-15, -855\right) \), \( \left(31, 778\right) \), \( \left(31, -809\right) \), \( \left(54, 801\right) \), \( \left(54, -855\right) \), \( \left(105, 1185\right) \), \( \left(105, -1290\right) \), \( \left(110, 1245\right) \), \( \left(110, -1355\right) \), \( \left(139, 1651\right) \), \( \left(139, -1790\right) \), \( \left(215, 3055\right) \), \( \left(215, -3270\right) \), \( \left(560, 12945\right) \), \( \left(560, -13505\right) \), \( \left(1480, 56185\right) \), \( \left(1480, -57665\right) \), \( \left(4884, 338901\right) \), \( \left(4884, -343785\right) \), \( \left(257340, 130416855\right) \), \( \left(257340, -130674195\right) \)
Invariants
Conductor: | \( 417450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-209513458687500 $ | = | $-1 \cdot 2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 11^{3} \cdot 23^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{178453547}{10074276} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-2} \cdot 23^{-4} \cdot 563^{3}$ | 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: | $1.4278571259125893998902624899\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.023664351495946576574396928794\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0793172916157505\dots$ | |||
Szpiro ratio: | $3.1247857904795246\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.60036505243033889573086130664\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.46553161423349878812921597831\dots$ | 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: | $ 128 $ = $ 2\cdot2\cdot2^{2}\cdot2\cdot2^{2} $ | 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! $ ≈ $ 8.9436451835927936279738533717 $ | 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
$\displaystyle 8.943645184 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.465532 \cdot 0.600365 \cdot 128}{2^2} \approx 8.943645184$
Modular invariants
Modular form 417450.2.a.m
For more coefficients, see the Downloads section to the right.
Modular degree: | 1474560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$11$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
$23$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 89 & 4 \\ 46 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 52 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 101 & 34 \\ 32 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 129 & 4 \\ 128 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$5068800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 417450.m
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 16698.bj2, its twist by $5$.
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.