Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+32100x-2374000\)
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(homogenize, simplify) |
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\(y^2z=x^3+32100xz^2-2374000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+32100x-2374000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{152096}{169}, \frac{60385844}{2197}\right) \) | $11.341595873121057966731973537$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1977248:60385844:2197]\) | $11.341595873121057966731973537$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{152096}{169}, \frac{60385844}{2197}\right) \) | $11.341595873121057966731973537$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-4551572736000000$ | = | $-1 \cdot 2^{14} \cdot 3^{6} \cdot 5^{6} \cdot 29^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{19600688}{24389} \) | = | $2^{4} \cdot 29^{-3} \cdot 107^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6905852121547408948120206533$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.47211159904963366583941910681$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8742214421492837$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.313428662784763$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.341595873121057966731973537$ |
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| Real period: | $\Omega$ | ≈ | $0.23305453832474295114295058331$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2864207801520761940319623052 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.286420780 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.233055 \cdot 11.341596 \cdot 2}{1^2} \\ & \approx 5.286420780\end{aligned}$$
Modular invariants
Modular form 417600.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2488320 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}^{*}$ | additive | -1 | 6 | 14 | 0 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $29$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1049 & 690 \\ 1050 & 689 \end{array}\right),\left(\begin{array}{rr} 206 & 2235 \\ 3055 & 2791 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1739 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 3475 & 6 \\ 3474 & 7 \end{array}\right),\left(\begin{array}{rr} 2783 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 2609 & 690 \\ 1215 & 2069 \end{array}\right),\left(\begin{array}{rr} 2579 & 690 \\ 1125 & 2069 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$1508651827200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $2$ | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| $3$ | additive | $6$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | 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 417600.n
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 116.b2, 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.