Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-54473x+1711350\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-54473xz^2+1711350z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-871563x+108654854\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 480249 \) | = | $3^{4} \cdot 7^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $9099512692305651$ | = | $3^{4} \cdot 7^{8} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{21609}{11} \) | = | $3^{2} \cdot 7^{4} \cdot 11^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7546522191472337470258387113$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1077729461781969588737833190$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8678596339606705$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3886075406283753$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.36276230308873410154248942208$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4510492123549364061699576883 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.451049212 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.362762 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.451049212\end{aligned}$$
Modular invariants
Modular form 480249.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2419200 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $7$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 label 44.2.0.a.1, level \( 44 = 2^{2} \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 43 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 23 & 3 \end{array}\right),\left(\begin{array}{rr} 43 & 2 \\ 42 & 3 \end{array}\right),\left(\begin{array}{rr} 13 & 2 \\ 13 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[44])$ is a degree-$633600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/44\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $26$ | \( 9801 = 3^{4} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3969 = 3^{4} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 480249.h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 43659.e1, its twist by $-231$.
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 $0$.