Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-5384x+123120\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-5384xz^2+123120z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-436131x+88446114\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(98, 726)$ | $1.0633414539418407955850052780$ | $\infty$ |
| $(-23, 484)$ | $2.1070918640682449998684265138$ | $\infty$ |
Integral points
\((-70,\pm 390)\), \((-23,\pm 484)\), \((26,\pm 6)\), \((61,\pm 134)\), \((98,\pm 726)\), \((80605,\pm 22884430)\)
Invariants
| Conductor: | $N$ | = | \( 444048 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $3631275343872$ | = | $2^{15} \cdot 3^{2} \cdot 11^{4} \cdot 29^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{5011452097}{1054152} \) | = | $2^{-3} \cdot 3^{-2} \cdot 11^{-4} \cdot 29 \cdot 557^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1247812050633635589286342448$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.12958194716099408835247654872$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8998374598851433$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.8751343110039596$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2106209277549283338859545738$ |
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| Real period: | $\Omega$ | ≈ | $0.74570405418241880043216171821$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.187751904698799959276066868 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.187751905 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.745704 \cdot 2.210621 \cdot 8}{1^2} \\ & \approx 13.187751905\end{aligned}$$
Modular invariants
Modular form 444048.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 691200 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $29$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
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 |
|---|---|---|
| $2$ | 2G | 8.2.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.b.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
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 | $4$ | \( 841 = 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148016 = 2^{4} \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40368 = 2^{4} \cdot 3 \cdot 29^{2} \) |
| $29$ | additive | $170$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 444048.i consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 55506.n1, its twist by $-4$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.