Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-4575276005x+119118218323997\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-4575276005xz^2+119118218323997z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-73204416075x+7623492768319750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(39199, 30400\right)\) | \(\left(38859, 46720\right)\) |
$\hat{h}(P)$ | ≈ | $1.7843780579633479719042903343$ | $2.1058194364204956626057754850$ |
Torsion generators
\( \left(\frac{156171}{4}, -\frac{156175}{8}\right) \)
Integral points
\( \left(-75801, 5550400\right) \), \( \left(-75801, -5474600\right) \), \( \left(24159, 4750720\right) \), \( \left(24159, -4774880\right) \), \( \left(38859, 46720\right) \), \( \left(38859, -85580\right) \), \( \left(39063, -18560\right) \), \( \left(39063, -20504\right) \), \( \left(39199, 30400\right) \), \( \left(39199, -69600\right) \), \( \left(39595, 166228\right) \), \( \left(39595, -205824\right) \), \( \left(44895, 2026528\right) \), \( \left(44895, -2071424\right) \), \( \left(51699, 4530400\right) \), \( \left(51699, -4582100\right) \), \( \left(1470655, 1780881888\right) \), \( \left(1470655, -1782352544\right) \)
Invariants
Conductor: | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1187946541971000000000000 $ | = | $2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{7903870428425797297009}{886464000000} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-6} \cdot 19^{-1} \cdot 67^{3} \cdot 127^{3} \cdot 2341^{3}$ | 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: | $4.0440334518303273256924701034\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.7170532767515656401417914466\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0425519890633115\dots$ | |||
Szpiro ratio: | $6.051937747049002\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.5713843883410862562314653923\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.066936335495004888778689224134\dots$ | 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: | $ 384 $ = $ ( 2^{2} \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot2\cdot1 $ | 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! $ ≈ $ 22.949316825563690481382238549 $ | 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
$\displaystyle 22.949316826 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.066936 \cdot 3.571384 \cdot 384}{2^2} \approx 22.949316826$
Modular invariants
Modular form 418950.2.a.je
For more coefficients, see the Downloads section to the right.
Modular degree: | 398131200 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$5$ | $4$ | $I_{6}^{*}$ | Additive | 1 | 2 | 12 | 6 |
$7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7980 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 4171 & 5698 \\ 756 & 7979 \end{array}\right),\left(\begin{array}{rr} 6383 & 2268 \\ 7518 & 5627 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 7930 & 7971 \end{array}\right),\left(\begin{array}{rr} 7969 & 12 \\ 7968 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 666 & 1057 \\ 665 & 3326 \end{array}\right),\left(\begin{array}{rr} 4390 & 3423 \\ 7413 & 5692 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 2279 & 0 \\ 0 & 7979 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[7980])$ is a degree-$5718756556800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7980\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 418950.je
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570.k2, its twist by $105$.
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.