This is a model for the modular curve $X_0(24)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-4x+4\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-4xz^2+4z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-351x+1890\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1, 0)$ | $0$ | $2$ |
| $(4, 6)$ | $0$ | $4$ |
Integral points
\( \left(-2, 0\right) \), \((0,\pm 2)\), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \((4,\pm 6)\)
Invariants
| Conductor: | $N$ | = | \( 24 \) | = | $2^{3} \cdot 3$ |
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| Discriminant: | $\Delta$ | = | $2304$ | = | $2^{8} \cdot 3^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{35152}{9} \) | = | $2^{4} \cdot 3^{-2} \cdot 13^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.64535228607985727265901067683$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1074504064531541456038320911$ |
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| $abc$ quality: | $Q$ | ≈ | $0.972547111469975$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.038496858107288$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $4.3130312949992864708773499976$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.53912891187491080885966874970 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.539128912 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.313031 \cdot 1.000000 \cdot 8}{8^2} \\ & \approx 0.539128912\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2Cs | 8.96.0.42 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.192.1-24.bu.1.7, level \( 24 = 2^{3} \cdot 3 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 22 & 3 \end{array}\right),\left(\begin{array}{rr} 17 & 8 \\ 16 & 9 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 6 & 19 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 20 & 21 \end{array}\right),\left(\begin{array}{rr} 17 & 0 \\ 16 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 8 = 2^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 24.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.2985984.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.181398528.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.36520347436056576.1 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.8.2393397489569403764736.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | 16.0.364791569817010176.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
We only show fields where the torsion growth is primitive.
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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
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 $S_4 \times \Z/2\Z$.