Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-26222934864x-1634449117709216\)
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(homogenize, simplify) |
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\(y^2z=x^3-26222934864xz^2-1634449117709216z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-26222934864x-1634449117709216\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 404928 \) | = | $2^{6} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-7159593201762656410991640576$ | = | $-1 \cdot 2^{14} \cdot 3^{14} \cdot 19 \cdot 37^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{166962959078001445737309395968}{599433319281236638491} \) | = | $-1 \cdot 2^{10} \cdot 3^{-8} \cdot 7^{6} \cdot 19^{-1} \cdot 37^{-10} \cdot 317^{3} \cdot 35171^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5625405778491819982666895470$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.2045627228618576249156294535$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0868469630613313$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.473573127242493$ | |||
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: | $\Omega$ | ≈ | $0.0059319726692948211708928154372$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 20 $ = $ 1\cdot2\cdot1\cdot( 2 \cdot 5 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.11863945338589642341785630874 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.118639453 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.005932 \cdot 1.000000 \cdot 20}{1^2} \\ & \approx 0.118639453\end{aligned}$$
Modular invariants
Modular form 404928.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1320222720 |
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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 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 |
| $3$ | $2$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
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 38.2.0.a.1, level \( 38 = 2 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 37 & 2 \\ 36 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 21 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 37 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[38])$ is a degree-$369360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38\Z)$.
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 | $2$ | \( 171 = 3^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 44992 = 2^{6} \cdot 19 \cdot 37 \) |
| $5$ | good | $2$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 21312 = 2^{6} \cdot 3^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 404928i consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 16872a1, 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 $0$.