Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-17034726259173x-27061436852750306309\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-17034726259173xz^2-27061436852750306309z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-272555620146768x-1731931958576019603760\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 423801 \) | = | $3^{2} \cdot 7^{2} \cdot 31^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-29899138071502687532752117055187$ | = | $-1 \cdot 3^{17} \cdot 7^{10} \cdot 31^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{7776720357545683677184}{425329947} \) | = | $-1 \cdot 2^{12} \cdot 3^{-11} \cdot 7^{-4} \cdot 31^{2} \cdot 179^{3} \cdot 701^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.9559602110902094227705100048$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5720429884908760529125740775$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1001959515897402$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.950318361375821$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.0011749943255467681768614494739$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $11.749943255467681768614494739 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $625$ = $25^2$ (exact) |
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BSD formula
$$\begin{aligned} 11.749943255 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{625 \cdot 0.001175 \cdot 1.000000 \cdot 16}{1^2} \\ & \approx 11.749943255\end{aligned}$$
Modular invariants
Modular form 423801.2.a.ci
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23569920000 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{11}^{*}$ | additive | -1 | 2 | 17 | 11 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $31$ | $1$ | $II^{*}$ | additive | -1 | 2 | 10 | 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 label 6.2.0.a.1, level \( 6 = 2 \cdot 3 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 5 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 47089 = 7^{2} \cdot 31^{2} \) |
| $7$ | additive | $32$ | \( 8649 = 3^{2} \cdot 31^{2} \) |
| $31$ | additive | $212$ | \( 441 = 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 423801.ci consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 20181.b1, its twist by $-651$.
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 $0$.