Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-329081x-71369485\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-329081xz^2-71369485z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5265291x-4572912314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1461/4, 1457/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 400554 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $75287160252178182$ | = | $2 \cdot 3^{10} \cdot 7^{4} \cdot 11 \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{223980311017}{4278582} \) | = | $2^{-1} \cdot 3^{-4} \cdot 7^{-4} \cdot 11^{-1} \cdot 6073^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0315734117909525704105243241$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.065660595428789684588134396700$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9368058527651448$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8545310198939227$ | |||
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: | $\Omega$ | ≈ | $0.19956200233321259849831710964$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1929920373314015759730737542 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.192992037 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.199562 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 3.192992037\end{aligned}$$
Modular invariants
Modular form 400554.2.a.de
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31416 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 25871 & 0 \\ 0 & 31415 \end{array}\right),\left(\begin{array}{rr} 10471 & 22168 \\ 21556 & 25839 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31409 & 8 \\ 31408 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 31410 & 31411 \end{array}\right),\left(\begin{array}{rr} 8977 & 9248 \\ 24820 & 5577 \end{array}\right),\left(\begin{array}{rr} 3928 & 28883 \\ 24259 & 9504 \end{array}\right),\left(\begin{array}{rr} 6240 & 5321 \\ 10387 & 17766 \end{array}\right),\left(\begin{array}{rr} 22016 & 7395 \\ 21845 & 12938 \end{array}\right)$.
The torsion field $K:=\Q(E[31416])$ is a degree-$3201968583475200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31416\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 28611 = 3^{2} \cdot 11 \cdot 17^{2} \) |
| $3$ | additive | $8$ | \( 44506 = 2 \cdot 7 \cdot 11 \cdot 17^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 57222 = 2 \cdot 3^{2} \cdot 11 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 36414 = 2 \cdot 3^{2} \cdot 7 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 400554de
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462c4, its twist by $-51$.
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 $0$.