Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-380x+1622\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-380xz^2+1622z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6075x+97750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 10)$ | $1.3000781548751447684360067618$ | $\infty$ |
| $(-21, 10)$ | $0$ | $2$ |
Integral points
\( \left(-21, 10\right) \), \( \left(4, 10\right) \), \( \left(4, -15\right) \), \( \left(18, 10\right) \), \( \left(18, -29\right) \), \( \left(175, 2208\right) \), \( \left(175, -2384\right) \)
Invariants
| Conductor: | $N$ | = | \( 20475 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $2495390625$ | = | $3^{3} \cdot 5^{7} \cdot 7 \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{14348907}{5915} \) | = | $3^{15} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{-2}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.50054112562504937588305710798$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.57883090275902823426613386786$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9497565867678647$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.964819938959719$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3000781548751447684360067618$ |
|
| Real period: | $\Omega$ | ≈ | $1.3106651549715118711663789824$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.4079342726690176591995169747 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.407934273 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.310665 \cdot 1.300078 \cdot 8}{2^2} \\ & \approx 3.407934273\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10752 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 3644 & 1 \\ 1819 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4682 & 1 \\ 3119 & 0 \end{array}\right),\left(\begin{array}{rr} 1369 & 4096 \\ 1364 & 4095 \end{array}\right),\left(\begin{array}{rr} 1094 & 1 \\ 2183 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 5457 & 4 \\ 5456 & 5 \end{array}\right),\left(\begin{array}{rr} 4201 & 4 \\ 2942 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$9738607656960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 20475b
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4095f1, its twist by $-15$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2555280.4 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.7998583451040000.79 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | add | add | nonsplit | ord | nonsplit | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | - | - | 1 | 3 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.