Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-175x+789\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-175xz^2+789z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-226827x+40222278\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9, 2)$ | $0$ | $5$ |
Integral points
\( \left(9, 2\right) \), \( \left(9, -12\right) \), \( \left(23, 86\right) \), \( \left(23, -110\right) \)
Invariants
| Conductor: | $N$ | = | \( 574 \) | = | $2 \cdot 7 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $22050784$ | = | $2^{5} \cdot 7^{5} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( \frac{592915705201}{22050784} \) | = | $2^{-5} \cdot 7^{-5} \cdot 31^{3} \cdot 41^{-1} \cdot 271^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.17802638656100223425188153924$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.17802638656100223425188153924$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9033608159542095$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.267259489716165$ | |||
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: | $\Omega$ | ≈ | $2.1296506457988495940000575462$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 25 $ = $ 5\cdot5\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $5$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.1296506457988495940000575462 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.129650646 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.129651 \cdot 1.000000 \cdot 25}{5^2} \\ & \approx 2.129650646\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 120 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $41$ | $1$ | $I_{1}$ | split 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 \( 11480 = 2^{3} \cdot 5 \cdot 7 \cdot 41 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 2871 & 10 \\ 2875 & 51 \end{array}\right),\left(\begin{array}{rr} 2871 & 5750 \\ 0 & 5167 \end{array}\right),\left(\begin{array}{rr} 1641 & 10 \\ 8205 & 51 \end{array}\right),\left(\begin{array}{rr} 5741 & 10 \\ 5745 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 11425 & 11361 \end{array}\right),\left(\begin{array}{rr} 11471 & 10 \\ 11470 & 11 \end{array}\right),\left(\begin{array}{rr} 3081 & 10 \\ 3925 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[11480])$ is a degree-$85316861952000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11480\Z)$.
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 | $4$ | \( 287 = 7 \cdot 41 \) |
| $5$ | good | $2$ | \( 41 \) |
| $7$ | split multiplicative | $8$ | \( 82 = 2 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 14 = 2 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 574j
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 |
|---|---|---|---|
| $3$ | 3.3.2296.1 | \(\Z/10\Z\) | not in database |
| $6$ | 6.6.12103630336.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.2.2930969733552.1 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.1945771207112214793287128936767578125.2 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | 7 | 41 |
|---|---|---|---|---|---|
| Reduction type | split | ord | ord | split | split |
| $\lambda$-invariant(s) | 6 | 0 | 6 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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 $0$.