Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-420x+3109\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-420xz^2+3109z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-544347x+153226998\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(13, 1)$ | $0$ | $5$ |
Integral points
\( \left(13, 1\right) \), \( \left(13, -15\right) \), \( \left(29, 113\right) \), \( \left(29, -143\right) \)
Invariants
| Conductor: | $N$ | = | \( 158 \) | = | $2 \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $82837504$ | = | $2^{20} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{8194759433281}{82837504} \) | = | $2^{-20} \cdot 79^{-1} \cdot 20161^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.33739064779017481227445257639$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.33739064779017481227445257639$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9613126805385601$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.8733743817961725$ | |||
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: | $\Omega$ | ≈ | $1.9304530478859940993079629035$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 20 $ = $ ( 2^{2} \cdot 5 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $5$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.5443624383087952794463703228 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.544362438 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.930453 \cdot 1.000000 \cdot 20}{5^2} \\ & \approx 1.544362438\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48 |
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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 semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $79$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $5$ | 5B.1.1 | 5.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1580 = 2^{2} \cdot 5 \cdot 79 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1571 & 10 \\ 1570 & 11 \end{array}\right),\left(\begin{array}{rr} 791 & 10 \\ 0 & 1107 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 1525 & 1461 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 791 & 10 \\ 795 & 51 \end{array}\right),\left(\begin{array}{rr} 161 & 10 \\ 805 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[1580])$ is a degree-$36912844800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1580\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 79 \) |
| $5$ | good | $2$ | \( 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 158c
consists of 2 curves linked by isogenies of
degree 5.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.316.1 | \(\Z/10\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.2.16826434992.1 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.70239841952395686029466273398468017578125.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 79 |
|---|---|---|---|---|
| Reduction type | split | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 8 | 0 | 2 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.