Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-185463947360x+31704893361448704\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-185463947360xz^2+31704893361448704z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15022579736187x+23112822192756896682\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12842037039478231349683920/67710253278978208441, 32257154407916237435838926220287459328/557162216599394100313157561261)$ | $42.928864096444045533589725405$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 444048 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-25964477306548469914312002540404736$ | = | $-1 \cdot 2^{35} \cdot 3^{5} \cdot 11^{8} \cdot 29^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{11872994862835724607989}{436955947638718464} \) | = | $-1 \cdot 2^{-23} \cdot 3^{-5} \cdot 11^{-8} \cdot 22813229^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.3745618100823414359021762109$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1559427570325406060974900652$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0544733032417744$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.883711519585418$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $42.928864096444045533589725405$ |
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| Real period: | $\Omega$ | ≈ | $0.011827082766286630234866864878$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.1235716597010278913947167254 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.123571660 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.011827 \cdot 42.928864 \cdot 16}{1^2} \\ & \approx 8.123571660\end{aligned}$$
Modular invariants
Modular form 444048.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3012065280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{27}^{*}$ | additive | -1 | 4 | 35 | 23 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $11$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $29$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 696 = 2^{3} \cdot 3 \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 349 & 2 \\ 349 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 695 & 2 \\ 694 & 3 \end{array}\right),\left(\begin{array}{rr} 233 & 2 \\ 233 & 3 \end{array}\right),\left(\begin{array}{rr} 175 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 553 & 2 \\ 553 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 695 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[696])$ is a degree-$25144197120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/696\Z)$.
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 | $4$ | \( 87 = 3 \cdot 29 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 148016 = 2^{4} \cdot 11 \cdot 29^{2} \) |
| $5$ | good | $2$ | \( 148016 = 2^{4} \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40368 = 2^{4} \cdot 3 \cdot 29^{2} \) |
| $29$ | additive | $254$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 444048.y consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 55506.g1, its twist by $-116$.
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.