Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-48165613x+96774410531\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-48165613xz^2+96774410531z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-62422634475x+4516043237259750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(2041, 82500\right)\) | \(\left(2215, 29892\right)\) |
$\hat{h}(P)$ | ≈ | $2.3800211323875338238052926978$ | $5.2026331178917269189509026531$ |
Torsion generators
\( \left(-7785, 3892\right) \)
Integral points
\( \left(-7785, 3892\right) \), \( \left(-7479, 200548\right) \), \( \left(-7479, -193070\right) \), \( \left(2041, 82500\right) \), \( \left(2041, -84542\right) \), \( \left(2215, 29892\right) \), \( \left(2215, -32108\right) \), \( \left(5815, 112692\right) \), \( \left(5815, -118508\right) \), \( \left(41499, 8320012\right) \), \( \left(41499, -8361512\right) \)
Invariants
Conductor: | \( 476850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $3103920976838501952000000 $ | = | $2^{12} \cdot 3^{7} \cdot 5^{6} \cdot 11 \cdot 17^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{32765849647039657}{8229948198912} \) | = | $2^{-12} \cdot 3^{-7} \cdot 11^{-1} \cdot 17^{-4} \cdot 319993^{3}$ | 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: | $3.4095133922500087732295649052\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.1881877640048505458044179296\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9895849394031041\dots$ | |||
Szpiro ratio: | $4.947172635136613\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $11.651771365970535605388737037\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.074892957501928678157338639453\dots$ | 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: | $ 96 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot2\cdot1\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! $ ≈ $ 20.943254825611698870587199084 $ | 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
$\displaystyle 20.943254826 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.074893 \cdot 11.651771 \cdot 96}{2^2} \approx 20.943254826$
Modular invariants
Modular form 476850.2.a.gx
For more coefficients, see the Downloads section to the right.
Modular degree: | 99090432 | 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 |
$3$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$5$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$11$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$17$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 1636 & 13465 \\ 13895 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7919 & 4480 \\ 260 & 17919 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13463 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 1691 & 15150 \\ 2810 & 5051 \end{array}\right),\left(\begin{array}{rr} 19456 & 17955 \\ 14965 & 13466 \end{array}\right),\left(\begin{array}{rr} 20761 & 20760 \\ 9550 & 11791 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$762373472256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 476850.gx
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.a4, its twist by $85$.
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.