Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-108x+5750352\)
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(homogenize, simplify) |
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\(y^2z=x^3-108xz^2+5750352z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108x+5750352\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 411840 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-14284748708904960$ | = | $-1 \cdot 2^{23} \cdot 3^{9} \cdot 5 \cdot 11^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{27}{2768480} \) | = | $-1 \cdot 2^{-5} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-3} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7787133677088084217804140360$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.084966619632191810891868073881$ |
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| $abc$ quality: | $Q$ | ≈ | $1.3040355200506624$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4538475472748478$ | |||
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: | $\Omega$ | ≈ | $0.31438346851221268675089161324$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.2575338740488507470035664530 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.257533874 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.314383 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.257533874\end{aligned}$$
Modular invariants
Modular form 411840.2.a.bf
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2764800 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{13}^{*}$ | additive | -1 | 6 | 23 | 5 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 17159 & 0 \end{array}\right),\left(\begin{array}{rr} 10297 & 2 \\ 10297 & 3 \end{array}\right),\left(\begin{array}{rr} 2641 & 2 \\ 2641 & 3 \end{array}\right),\left(\begin{array}{rr} 12871 & 2 \\ 12871 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8581 & 2 \\ 8581 & 3 \end{array}\right),\left(\begin{array}{rr} 11441 & 2 \\ 11441 & 3 \end{array}\right),\left(\begin{array}{rr} 7801 & 2 \\ 7801 & 3 \end{array}\right),\left(\begin{array}{rr} 17159 & 2 \\ 17158 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$6121410527232000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $4$ | \( 2145 = 3 \cdot 5 \cdot 11 \cdot 13 \) |
| $3$ | additive | $2$ | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 82368 = 2^{6} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 37440 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 31680 = 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 411840bf consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12870d1, 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 $0$.