Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-1517824910x+22759806218315\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-1517824910xz^2+22759806218315z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1967101083387x+1061911025437963734\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(22493, -11177\right)\) | \(\left(22583, 12173\right)\) |
$\hat{h}(P)$ | ≈ | $1.2352457927090264796036260845$ | $1.7644126470131876171008376102$ |
Torsion generators
\( \left(\frac{89971}{4}, -\frac{89975}{8}\right) \)
Integral points
\( \left(22493, -11177\right) \), \( \left(22493, -11317\right) \), \( \left(22523, -3397\right) \), \( \left(22523, -19127\right) \), \( \left(22583, 12173\right) \), \( \left(22583, -34757\right) \), \( \left(22873, 87623\right) \), \( \left(22873, -110497\right) \), \( \left(23305, 200615\right) \), \( \left(23305, -223921\right) \), \( \left(24943, 635483\right) \), \( \left(24943, -660427\right) \), \( \left(33413, 3040963\right) \), \( \left(33413, -3074377\right) \), \( \left(47933, 7730923\right) \), \( \left(47933, -7778857\right) \), \( \left(60713, 12395623\right) \), \( \left(60713, -12456337\right) \), \( \left(133049, 46581895\right) \), \( \left(133049, -46714945\right) \)
Invariants
Conductor: | \( 482790 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $543010005742512000 $ | = | $2^{7} \cdot 3 \cdot 5^{3} \cdot 7^{2} \cdot 11^{6} \cdot 19^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{218289391029690300712901881}{306514992000} \) | = | $2^{-7} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{-2} \cdot 19^{-4} \cdot 602112361^{3}$ | 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: | $3.5678947142436943433699070487\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.3689470778445090713389352597\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0308751752376573\dots$ | |||
Szpiro ratio: | $5.733422713482888\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.1358562663728684869140006575\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13165334041246727684325946487\dots$ | 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: | $ 336 $ = $ 7\cdot1\cdot3\cdot2\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! $ ≈ $ 23.620179417146645726601848358 $ | 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
$\displaystyle 23.620179417 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.131653 \cdot 2.135856 \cdot 336}{2^2} \approx 23.620179417$
Modular invariants
Modular form 482790.2.a.fw
For more coefficients, see the Downloads section to the right.
Modular degree: | 137625600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$11$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $4$ | $I_{4}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25080 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 25073 & 8 \\ 25072 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3136 & 17963 \\ 16819 & 12288 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 22441 & 2288 \\ 3124 & 9153 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 25074 & 25075 \end{array}\right),\left(\begin{array}{rr} 16248 & 24233 \\ 13387 & 15654 \end{array}\right),\left(\begin{array}{rr} 1376 & 22803 \\ 4565 & 15962 \end{array}\right),\left(\begin{array}{rr} 15959 & 0 \\ 0 & 25079 \end{array}\right),\left(\begin{array}{rr} 14444 & 15961 \\ 24343 & 4566 \end{array}\right)$.
The torsion field $K:=\Q(E[25080])$ is a degree-$1198215659520000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25080\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 482790.fw
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3990.g1, 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.