Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+731445x+143319418\)
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(homogenize, simplify) |
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\(y^2z=x^3+731445xz^2+143319418z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+731445x+143319418\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(581, 27648)$ | $3.2950736451086548482001028011$ | $\infty$ |
| $(-187, 0)$ | $0$ | $2$ |
Integral points
\( \left(-187, 0\right) \), \((581,\pm 27648)\), \((902,\pm 39204)\)
Invariants
| Conductor: | $N$ | = | \( 400752 \) | = | $2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-33918705435455520768$ | = | $-1 \cdot 2^{20} \cdot 3^{8} \cdot 11^{8} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{8181353375}{6412032} \) | = | $2^{-8} \cdot 3^{-2} \cdot 5^{3} \cdot 11^{-2} \cdot 13^{3} \cdot 23^{-1} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4355280315601828873458771684$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.0058729297330025397999493605025$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8955031105784524$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.0401161755225665$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.2950736451086548482001028011$ |
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| Real period: | $\Omega$ | ≈ | $0.13305169809688950658073561511$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5073211506881118533657554092 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.507321151 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.133052 \cdot 3.295074 \cdot 32}{2^2} \\ & \approx 3.507321151\end{aligned}$$
Modular invariants
Modular form 400752.2.a.cu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8847360 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $23$ | $1$ | $I_{1}$ | split 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 276 = 2^{2} \cdot 3 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 185 & 4 \\ 94 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 194 & 1 \\ 179 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 273 & 4 \\ 272 & 5 \end{array}\right),\left(\begin{array}{rr} 73 & 208 \\ 68 & 207 \end{array}\right)$.
The torsion field $K:=\Q(E[276])$ is a degree-$102592512$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/276\Z)$.
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 | $2$ | \( 25047 = 3^{2} \cdot 11^{2} \cdot 23 \) |
| $3$ | additive | $8$ | \( 44528 = 2^{4} \cdot 11^{2} \cdot 23 \) |
| $11$ | additive | $72$ | \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 400752cu
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1518m1, its twist by $-132$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.