Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+32928x-3413536\)
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(homogenize, simplify) |
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\(y^2z=x^3+32928xz^2-3413536z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+32928x-3413536\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 423360 \) | = | $2^{6} \cdot 3^{3} \cdot 5 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-7318708992000000$ | = | $-1 \cdot 2^{14} \cdot 3^{5} \cdot 5^{6} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{8429568}{15625} \) | = | $2^{13} \cdot 3 \cdot 5^{-6} \cdot 7^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7287971662846253387531821005$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51058473917467987953428392831$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0691763773828689$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.365987772103896$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.21895941380738345970902167729$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot3\cdot( 2 \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.9412694485329022747623901913 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.941269449 \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 0.218959 \cdot 1.000000 \cdot 18}{1^2} \\ & \approx 3.941269449\end{aligned}$$
Modular invariants
Modular form 423360.2.a.wb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2612736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 6 | 14 | 0 |
| $3$ | $3$ | $IV$ | additive | 1 | 3 | 5 | 0 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-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 \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 48 & 119 \\ 119 & 48 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 95 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 163 & 6 \\ 162 & 7 \end{array}\right),\left(\begin{array}{rr} 83 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 148 & 147 \\ 147 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 69 & 140 \\ 70 & 139 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$9289728$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 | $2$ | \( 1323 = 3^{3} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 224 = 2^{5} \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 84672 = 2^{6} \cdot 3^{3} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 423360.wb
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 540.b2, its twist by $-168$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.