Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+9408x-455504\)
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(homogenize, simplify) |
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\(y^2z=x^3+9408xz^2-455504z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9408x-455504\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-142926254714880$ | = | $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5 \cdot 7^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{7077888}{10985} \) | = | $2^{18} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4013876416290231610013112070$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53936768562560622381740859541$ |
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| $abc$ quality: | $Q$ | ≈ | $1.001928731766074$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0363149533185547$ | |||
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: | $\Omega$ | ≈ | $0.30683138776143129050300429732$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.61366277552286258100600859465 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.613662776 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.306831 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.613662776\end{aligned}$$
Modular invariants
Modular form 458640.2.a.iu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1306368 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 5459 \end{array}\right),\left(\begin{array}{rr} 5455 & 6 \\ 5454 & 7 \end{array}\right),\left(\begin{array}{rr} 3277 & 3906 \\ 861 & 799 \end{array}\right),\left(\begin{array}{rr} 3186 & 3451 \\ 133 & 3984 \end{array}\right),\left(\begin{array}{rr} 1175 & 1554 \\ 1176 & 1553 \end{array}\right),\left(\begin{array}{rr} 4201 & 3906 \\ 903 & 799 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 2729 & 0 \\ 0 & 5459 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$7303955742720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $2$ | \( 9555 = 3 \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 458640iu
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 585b1, its twist by $28$.
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 $0$.