Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+737844x+2359998704\)
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(homogenize, simplify) |
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\(y^2z=x^3+737844xz^2+2359998704z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+737844x+2359998704\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1148, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1148, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 404928 \) | = | $2^{6} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-2431772873083868479488$ | = | $-1 \cdot 2^{16} \cdot 3^{10} \cdot 19^{8} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( \frac{929843593713212}{50899738433877} \) | = | $2^{2} \cdot 3^{-4} \cdot 19^{-8} \cdot 37^{-1} \cdot 61487^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7832546122204166415693102102$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3097522271397680499820447631$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9854979810419444$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.390439141953552$ | |||
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: | $\Omega$ | ≈ | $0.11029197224533628315062931884$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.88233577796269026520503455074 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.882335778 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.110292 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.882335778\end{aligned}$$
Modular invariants
Modular form 404928.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13631488 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | 1 | 6 | 16 | 0 |
| $3$ | $2$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $19$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.24.0.91 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 33744 = 2^{4} \cdot 3 \cdot 19 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 33646 & 33731 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 33729 & 16 \\ 33728 & 17 \end{array}\right),\left(\begin{array}{rr} 33731 & 11232 \\ 25044 & 24983 \end{array}\right),\left(\begin{array}{rr} 11560 & 22497 \\ 32607 & 10 \end{array}\right),\left(\begin{array}{rr} 22495 & 0 \\ 0 & 33743 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 4234 & 14067 \\ 7263 & 28222 \end{array}\right),\left(\begin{array}{rr} 30193 & 22512 \\ 16584 & 11377 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 33740 & 33741 \end{array}\right)$.
The torsion field $K:=\Q(E[33744])$ is a degree-$1378383723233280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/33744\Z)$.
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 | $2$ | \( 333 = 3^{2} \cdot 37 \) |
| $3$ | additive | $8$ | \( 44992 = 2^{6} \cdot 19 \cdot 37 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 21312 = 2^{6} \cdot 3^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 10944 = 2^{6} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 404928bc
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 16872e4, 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 $0$.