Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-640889x-197533063\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-640889xz^2-197533063z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-830592171x-9213610810842\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 874 \) | = | $2 \cdot 19 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $20192896$ | = | $2^{7} \cdot 19^{3} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{29112011033527546515217}{20192896} \) | = | $2^{-7} \cdot 17^{3} \cdot 19^{-3} \cdot 23^{-1} \cdot 43^{3} \cdot 42083^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6152053552335871159684604563$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6152053552335871159684604563$ |
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| $abc$ quality: | $Q$ | ≈ | $1.014697315204348$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.636914804393003$ | |||
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: | $\Omega$ | ≈ | $0.16873448361931716149071537806$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 21 $ = $ 7\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5434241560056603913050229392 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.543424156 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.168734 \cdot 1.000000 \cdot 21}{1^2} \\ & \approx 3.543424156\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4032 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10488 = 2^{3} \cdot 3 \cdot 19 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 5245 & 6 \\ 5247 & 19 \end{array}\right),\left(\begin{array}{rr} 2623 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 553 & 6 \\ 1659 & 19 \end{array}\right),\left(\begin{array}{rr} 10033 & 6 \\ 9123 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 10483 & 6 \\ 10482 & 7 \end{array}\right),\left(\begin{array}{rr} 8306 & 2187 \\ 3943 & 5251 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10488])$ is a degree-$151574280929280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10488\Z)$.
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 | $4$ | \( 437 = 19 \cdot 23 \) |
| $3$ | good | $2$ | \( 46 = 2 \cdot 23 \) |
| $7$ | good | $2$ | \( 437 = 19 \cdot 23 \) |
| $19$ | split multiplicative | $20$ | \( 46 = 2 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 38 = 2 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 874.f
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.3496.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.6348.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.42728167936.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.120891312.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.329994432.6 | \(\Z/6\Z\) | not in database |
| $9$ | 9.3.5165464281139372032.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.338360565652250330668335973366267800029952.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.720414573472620615613109468777202843648.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.538787279963592108834499842208640896025493504.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 19 | 23 |
|---|---|---|---|---|---|---|
| Reduction type | split | ord | ord | ord | split | split |
| $\lambda$-invariant(s) | 1 | 2 | 0 | 2 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.