This is a model for the modular curve $X_0(24)$.
Minimal Weierstrass equation
\(y^2=x^3-x^2-4x+4\) 
Mordell-Weil group structure
\(\Z/{2}\Z \times \Z/{4}\Z\)
Torsion generators
\( \left(1, 0\right) \), \( \left(4, 6\right) \)
Integral points
\( \left(-2, 0\right) \), \((0,\pm 2)\), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \((4,\pm 6)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 24 \) | = | \(2^{3} \cdot 3\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(2304 \) | = | \(2^{8} \cdot 3^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{35152}{9} \) | = | \(2^{4} \cdot 3^{-2} \cdot 13^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | \(-0.64535228607985727265901067683\dots\) | ||
| Stable Faltings height: | \(-1.1074504064531541456038320911\dots\) | ||
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: | \(4.3130312949992864708773499976\dots\) | ||
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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: | \( 8 \) = \( 2^{2}\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(8\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
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: | 1 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.53912891187491080885966874970005048289 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 3 | 8 | 0 |
| \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 X190c.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 0 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 5 \end{array}\right)$ and has index 96.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | Cs |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 |
|---|---|---|
| Reduction type | add | nonsplit |
| $\lambda$-invariant(s) | - | 0 |
| $\mu$-invariant(s) | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 and 4.
Its isogeny class 24a
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \times \Z/8\Z\) | 4.4.2304.1-72.1-b4 |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.0.2985984.1 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.2.181398528.1 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | 16.0.36520347436056576.1 | \(\Z/8\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.8.2393397489569403764736.2 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
| $16$ | 16.0.364791569817010176.2 | \(\Z/2\Z \times \Z/16\Z\) | Not in database |
We only show fields where the torsion growth is primitive.
Additional information
This curve is also a quotient of the genus-3 hyperelliptic curve $$y^2 = x^8 + 14x^4 + 1$$ which has geometric automorphism group $S_4 \times \Z/2\Z$.