Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-14974804236x-766286848286192\)
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(homogenize, simplify) |
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\(y^2z=x^3-14974804236xz^2-766286848286192z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14974804236x-766286848286192\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 401472 \) | = | $2^{6} \cdot 3^{2} \cdot 17 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $-38755100354046815361170474532864$ | = | $-1 \cdot 2^{24} \cdot 3^{9} \cdot 17^{15} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( -\frac{1943299427371886688757286977}{202796948353367429302464} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-3} \cdot 7^{3} \cdot 17^{-15} \cdot 41^{-1} \cdot 79^{3} \cdot 2256601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.8003496942098127870184615086$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.2113227790358399771949907080$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0364691817074154$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.360457338163839$ | |||
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: | $\Omega$ | ≈ | $0.0067838092215540954942746946845$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 60 $ = $ 2\cdot2\cdot( 3 \cdot 5 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.40702855329324572965648168107 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.407028553 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.006784 \cdot 1.000000 \cdot 60}{1^2} \\ & \approx 0.407028553\end{aligned}$$
Modular invariants
Modular form 401472.2.a.en
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1224253440 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $17$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $41$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16728 = 2^{3} \cdot 3 \cdot 17 \cdot 41 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 13529 & 16722 \\ 15495 & 16709 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 8363 & 0 \\ 0 & 16727 \end{array}\right),\left(\begin{array}{rr} 3163 & 6 \\ 1125 & 19 \end{array}\right),\left(\begin{array}{rr} 12545 & 0 \\ 0 & 16727 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3488 & 699 \\ 1 & 2092 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 16723 & 6 \\ 16722 & 7 \end{array}\right),\left(\begin{array}{rr} 12551 & 16722 \\ 12552 & 16721 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[16728])$ is a degree-$994550847897600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16728\Z)$.
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 | $2$ | \( 6273 = 3^{2} \cdot 17 \cdot 41 \) |
| $3$ | additive | $6$ | \( 2624 = 2^{6} \cdot 41 \) |
| $5$ | good | $2$ | \( 23616 = 2^{6} \cdot 3^{2} \cdot 41 \) |
| $17$ | split multiplicative | $18$ | \( 23616 = 2^{6} \cdot 3^{2} \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 401472en
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4182h2, its twist by $24$.
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 $0$.