Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-1339169x-596933313\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-1339169xz^2-596933313z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108472716x-434838967056\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-669, 0)$ | $0$ | $2$ |
Integral points
\( \left(-669, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 55488 \) | = | $2^{6} \cdot 3 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $304200010432512$ | = | $2^{20} \cdot 3^{10} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{206226044828441}{236196} \) | = | $2^{-2} \cdot 3^{-10} \cdot 11^{3} \cdot 41^{3} \cdot 131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0634077797925157308337764296$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.31538367293854374664554459294$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0744894925088866$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.937446731441331$ | |||
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: | $\Omega$ | ≈ | $0.14034298538871381709358660441$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2^{2}\cdot( 2 \cdot 5 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.8068597077742763418717320882 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.806859708 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.140343 \cdot 1.000000 \cdot 80}{2^2} \\ & \approx 2.806859708\end{aligned}$$
Modular invariants
Modular form 55488.2.a.cu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 614400 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | 1 | 6 | 20 | 2 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $17$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
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 | 8.6.0.2 |
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 680 = 2^{3} \cdot 5 \cdot 17 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 440 & 331 \end{array}\right),\left(\begin{array}{rr} 258 & 5 \\ 375 & 82 \end{array}\right),\left(\begin{array}{rr} 339 & 674 \\ 0 & 679 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 511 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 661 & 20 \\ 660 & 21 \end{array}\right),\left(\begin{array}{rr} 137 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[680])$ is a degree-$200540160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/680\Z)$.
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 | $2$ | \( 17 \) |
| $3$ | split multiplicative | $4$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $5$ | good | $2$ | \( 18496 = 2^{6} \cdot 17^{2} \) |
| $17$ | additive | $82$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 55488.cu
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 1734.d2, its twist by $8$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.314432.3 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.39304000.2 | \(\Z/10\Z\) | not in database |
| $8$ | 8.4.32033064370176.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.98867482624.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1544804416000000.24 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.4.192093959274237222717489152000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 17 |
|---|---|---|---|---|
| Reduction type | add | split | ord | add |
| $\lambda$-invariant(s) | - | 1 | 0 | - |
| $\mu$-invariant(s) | - | 0 | 1 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.