Minimal Weierstrass equation
\( y^2 + y = x^{3} + x^{2} - 104207741 x + 135268278965 \)
Mordell-Weil group structure
Infinite order Mordell-Weil generator and height
\(P\) | = | \( \left(-7253, 713854\right) \) |
\(\hat{h}(P)\) | ≈ | 8.61394688704 |
Integral points
\( \left(-7253, 713854\right) \)
Invariants
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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Conductor: | \( 10005 \) | = | \(3 \cdot 5 \cdot 23 \cdot 29\) | ||
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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Discriminant: | \(64514985611316331088611125 \) | = | \(3^{7} \cdot 5^{3} \cdot 23 \cdot 29^{13} \) | ||
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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j-invariant: | \( \frac{125147927114815865709295304704}{64514985611316331088611125} \) | = | \(2^{21} \cdot 3^{-7} \cdot 5^{-3} \cdot 23^{-1} \cdot 29^{-13} \cdot 71^{3} \cdot 79^{3} \cdot 6967^{3}\) | ||
Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
Sato-Tate Group: | $\mathrm{SU}(2)$ |
BSD invariants
magma: Rank(E);
sage: E.rank()
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Rank: | \(1\) | ||
magma: Regulator(E);
sage: E.regulator()
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Regulator: | \(8.61394688704\) | ||
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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Real period: | \(0.054677784008\) | ||
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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Tamagawa product: | \( 7 \) = \( 7\cdot1\cdot1\cdot1 \) | ||
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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Torsion order: | \(1\) | ||
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 10005.2.a.g
magma: ModularDegree(E);
sage: E.modular_degree()
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Modular degree: | 2140320 | ||
\( \Gamma_0(N) \)-optimal: | yes | ||
Manin constant: | 1 |
Special L-value
\( L'(E,1) \) ≈ \( 3.29694069142 \)
Local data
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(3\) | \(7\) | \( I_{7} \) | Split multiplicative | -1 | 1 | 7 | 7 |
\(5\) | \(1\) | \( I_{3} \) | Non-split multiplicative | 1 | 1 | 3 | 3 |
\(23\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
\(29\) | \(1\) | \( I_{13} \) | Non-split multiplicative | 1 | 1 | 13 | 13 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .
$p$-adic data
$p$-adic regulators
Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ss | split | nonsplit | ordinary | ordinary | ss | ordinary | ordinary | split | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary |
$\lambda$-invariant(s) | 8,1 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0,0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has no rational isogenies. Its isogeny class 10005.g consists of this curve only.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
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3 | 3.3.40020.1 | \(\Z/2\Z\) | Not in database |
6 | 6.6.16024012002000.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.