Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6127500x-5844490000\)
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(homogenize, simplify) |
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\(y^2z=x^3-6127500xz^2-5844490000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6127500x-5844490000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4423925}{961}, \frac{7507309025}{29791}\right) \) | $11.743399322998572606845529273$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([137141675:7507309025:29791]\) | $11.743399322998572606845529273$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4423925}{961}, \frac{7507309025}{29791}\right) \) | $11.743399322998572606845529273$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-32143520563200000000$ | = | $-1 \cdot 2^{27} \cdot 3^{6} \cdot 5^{8} \cdot 29^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{340836570625}{430592} \) | = | $-1 \cdot 2^{-9} \cdot 5^{4} \cdot 19^{3} \cdot 29^{-2} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6516136469938320911703934830$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.010371876469540968386916873132$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9926845309145379$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.520121568951014$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.743399322998572606845529273$ |
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| Real period: | $\Omega$ | ≈ | $0.047975094146336882292062887703$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot1\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7606882574266239254108692060 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.760688257 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047975 \cdot 11.743399 \cdot 12}{1^2} \\ & \approx 6.760688257\end{aligned}$$
Modular invariants
Modular form 417600.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18662400 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{17}^{*}$ | additive | 1 | 6 | 27 | 9 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $29$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.a.1.4, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 22 \\ 14 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 18 & 23 \\ 23 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 18 \\ 9 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 21 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ 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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 46400 = 2^{6} \cdot 5^{2} \cdot 29 \) |
| $5$ | additive | $10$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 417600l
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1450b1, its twist by $-120$.
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.