Minimal Weierstrass equation
\(y^2+xy+y=x^3-149756x-22313454\)
Mordell-Weil group structure
\(\Z^2 \times \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \( \left(-222, 51\right) \) | \( \left(\frac{2427}{4}, \frac{81583}{8}\right) \) |
| \(\hat{h}(P)\) | ≈ | $2.6880748497007697204274941212$ | $6.6914110691605371378526570389$ |
Torsion generators
\( \left(-\frac{905}{4}, \frac{901}{8}\right) \)
Integral points
\( \left(-222, 170\right) \), \( \left(-222, 51\right) \), \( \left(644, 11861\right) \), \( \left(644, -12506\right) \), \( \left(369630, 224539842\right) \), \( \left(369630, -224909473\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 118354 \) | = | \(2 \cdot 17 \cdot 59^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(97521393777992 \) | = | \(2^{3} \cdot 17^{2} \cdot 59^{6} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{8805624625}{2312} \) | = | \(2^{-3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{-2} \cdot 59^{3}\) |
| 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: | \(2\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(16.933549918261329525229680012\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.24269492060995119663753239133\) | ||
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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: | \( 4 \) = \( 1\cdot2\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 118354.2.a.b
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: | 835200 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 4.1096865530570789441841419297311223131 \)
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | 1 | 1 | 3 | 3 |
| \(17\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(59\) | \(2\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 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 X17.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ and has index 6.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
| \(3\) | B |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ordinary | ss | ordinary | ordinary | ordinary | nonsplit | ordinary | ss | ss | ordinary | ordinary | ordinary | ordinary | ss | add |
| $\lambda$-invariant(s) | 2 | 2 | 2,6 | 2 | 2 | 2 | 2 | 2 | 4,2 | 2,2 | 2 | 2 | 2 | 2 | 2,2 | - |
| $\mu$-invariant(s) | 1 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0 | 0 | 0 | 0,0 | - |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class 118354e
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-59}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-59})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $4$ | 4.0.32192288.1 | \(\Z/4\Z\) | Not in database |
| $6$ | 6.2.463143405393.2 | \(\Z/6\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $18$ | 18.0.95896334836435873520348644773754515456.1 | \(\Z/18\Z\) | 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.