Minimal Weierstrass equation
\(y^2+xy=x^3+x^2-3274066700x-72122050446000\)
Mordell-Weil group structure
trivial
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 303450 \) | = | \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-832357110436901683200000000 \) | = | \(-1 \cdot 2^{24} \cdot 3^{2} \cdot 5^{8} \cdot 7 \cdot 17^{10} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{4928752745352265}{1056964608} \) | = | \(-1 \cdot 2^{-24} \cdot 3^{-2} \cdot 5 \cdot 7^{-1} \cdot 17^{2} \cdot 15053^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.0099791164602677600062086220680\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 12 \) = \( 2\cdot2\cdot3\cdot1\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(16\) = $4^2$ (exact) | ||
Modular invariants
Modular form 303450.2.a.p
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 317260800 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 1.9159903603714099211920554370528193201 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{24}\) | Non-split multiplicative | 1 | 1 | 24 | 24 |
| \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(5\) | \(3\) | \(IV^{*}\) | Additive | -1 | 2 | 8 | 0 |
| \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(17\) | \(1\) | \(II^{*}\) | Additive | 1 | 2 | 10 | 0 |
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 \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(3\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class 303450.p
consists of 2 curves linked by isogenies of
degree 3.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-51}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.50575.2 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.17904814375.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $6$ | 6.2.125814104699495625.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.1174044256875.1 | \(\Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $18$ | 18.0.10192413563054697128187916638662805795771000000000000.2 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.2.97585226503891769403823258839684280523003123291015625.1 | \(\Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.