Minimal Weierstrass equation
\(y^2+xy+y=x^3-44501x+4223648\) 
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \(\left(\frac{503}{4}, \frac{5793}{8}\right)\) |
| \(\hat{h}(P)\) | ≈ | $4.9391319168317635869021964390$ |
Integral points
\(\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 277350 \) | = | \(2 \cdot 3 \cdot 5^{2} \cdot 43^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-2080125000000000 \) | = | \(-1 \cdot 2^{9} \cdot 3^{2} \cdot 5^{12} \cdot 43^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{337335507529}{72000000} \) | = | \(-1 \cdot 2^{-9} \cdot 3^{-2} \cdot 5^{-6} \cdot 43 \cdot 1987^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | \(1.6605656343134270820294530662\dots\) | ||
| Stable Faltings height: | \(0.22897999214744982415026631403\dots\) | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(4.9391319168317635869021964390\dots\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.44446385313115259733915104599\dots\) | ||
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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: | \( 4 \) = \( 1\cdot2\cdot2\cdot1 \) | ||
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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: | \(1\) (exact) | ||
Modular invariants
Modular form 277350.2.a.bu
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: | 1741824 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 8.7810624115124047044513445031671417004 \)
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(I_{9}\) | Non-split multiplicative | 1 | 1 | 9 | 9 |
| \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | -1 | 1 | 2 | 2 |
| \(5\) | \(2\) | \(I_6^{*}\) | Additive | 1 | 2 | 12 | 6 |
| \(43\) | \(1\) | \(II\) | Additive | -1 | 2 | 2 | 0 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X4.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.
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
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
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 277350.bu
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{645}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.14792.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.1750426112.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $6$ | 6.0.1085087693838375.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.2.31753823688000.3 | \(\Z/6\Z\) | 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 |
| $18$ | 18.6.59887446382919467856117858337712210990264892578125.2 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.334914874551552874275715410263066236540416000000000.1 | \(\Z/6\Z\) | 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.