Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2-19233x-1020257\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z-19233xz^2-1020257z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24926400x-47900214000\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 475 \) | = | $5^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-296875$ | = | $-1 \cdot 5^{6} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( -\frac{50357871050752}{19} \) | = | $-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.83815811320836799082047455020$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.033439156991317803520094883587$ |
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| $abc$ quality: | $Q$ | ≈ | $1.104947099482495$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.6858184004953$ | |||
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: | $\Omega$ | ≈ | $0.20270101314312401414002509743$ |
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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)$ | ≈ | $1.8243091182881161272602258769 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.824309118 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.202701 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.824309118\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 324 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | 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 | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5130 = 2 \cdot 3^{3} \cdot 5 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 4103 & 0 \\ 0 & 5129 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 4438 & 3499 \end{array}\right),\left(\begin{array}{rr} 5041 & 75 \\ 3705 & 4196 \end{array}\right),\left(\begin{array}{rr} 4741 & 2070 \\ 4435 & 121 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 5077 & 54 \\ 5076 & 55 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[5130])$ is a degree-$86164300800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5130\Z)$.
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 |
|---|---|---|---|
| $5$ | additive | $14$ | \( 19 \) |
| $19$ | split multiplicative | $20$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 475a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a2, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.146611125.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.320638530375.5 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.19494000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.788891024171390625.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.4659806991485780193672000000000.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.48743226988750074577830865531416000000000.1 | \(\Z/18\Z\) | 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 | ss | ord | add | ord | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 0,3 | 0 | - | 4 | 0 | 0 | 0 | 1 | 0,0 | 0 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | 0,0 | 2 | - | 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 $0$.