Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-223449x-40646908\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-223449xz^2-40646908z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3575187x-2604977298\)
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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: | $\Delta$ | = | $-1671339065933691$ | = | $-1 \cdot 3^{6} \cdot 7^{6} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{8120601}{11} \) | = | $-1 \cdot 3^{3} \cdot 11^{-1} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8263506852286491271082607239$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.89485817003224764317301005527$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8914148062977623$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7124702815590727$ | |||
| 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: | $\Omega$ | ≈ | $0.10978413821541106574630279444$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6348193171698655779112670666 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.634819317 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.109784 \cdot 1.000000 \cdot 24}{1^2} \\ & \approx 2.634819317\end{aligned}$$
Modular invariants
Modular form 480249.2.a.bo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2488320 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $3$ | $IV$ | additive | -1 | 4 | 6 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B | 7.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1385 & 1372 \\ 0 & 1187 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1373 & 14 \\ 1372 & 15 \end{array}\right),\left(\begin{array}{rr} 465 & 8 \\ 602 & 1349 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 251 & 1372 \\ 371 & 1287 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right)$.
The torsion field $K:=\Q(E[1386])$ is a degree-$6466521600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1386\Z)$.
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 | $6$ | \( 77 = 7 \cdot 11 \) |
| $7$ | additive | $26$ | \( 9801 = 3^{4} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3969 = 3^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 480249.bo
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 891.c1, its twist by $77$.
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 $0$.