Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-41757x-3273877\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-41757xz^2-3273877z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-668115x-210196242\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 5022 \) | = | $2 \cdot 3^{4} \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $-3661038$ | = | $-1 \cdot 2 \cdot 3^{10} \cdot 31 $ |
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| j-invariant: | $j$ | = | \( -\frac{136365617642625}{62} \) | = | $-1 \cdot 2^{-1} \cdot 3^{2} \cdot 5^{3} \cdot 7^{6} \cdot 31^{-1} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0352354867652679384212213787$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11972524620850986225851701460$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0622437550292627$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.108496964685501$ | |||
| 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: | $\Omega$ | ≈ | $0.16698857131736461298142264121$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.16698857131736461298142264121 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.166988571 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.166989 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.166988571\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11592 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $IV^{*}$ | additive | -1 | 4 | 10 | 0 |
| $31$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 \( 15624 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 31 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1016 & 7 \\ 3017 & 15618 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15615 & 15614 \\ 13790 & 15515 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 3907 & 7826 \\ 0 & 15067 \end{array}\right),\left(\begin{array}{rr} 11719 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15611 & 14 \\ 15610 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 7805 & 15618 \end{array}\right)$.
The torsion field $K:=\Q(E[15624])$ is a degree-$111967233638400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15624\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 2511 = 3^{4} \cdot 31 \) |
| $3$ | additive | $4$ | \( 62 = 2 \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 5022.b
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.20088.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.100074880512.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | \(\Q(\zeta_{9})\) | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.2617583593392.6 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.1774123964891215885014663168.1 | \(\Z/14\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ss | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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 $0$.