Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-23693x+144008\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-23693xz^2+144008z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-30705507x+6810965406\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(284, 3900)$ | $3.0991265818443107272533769092$ | $\infty$ |
| $(-157, 78)$ | $0$ | $2$ |
Integral points
\( \left(-157, 78\right) \), \( \left(284, 3900\right) \), \( \left(284, -4185\right) \)
Invariants
| Conductor: | $N$ | = | \( 19110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $841958739148800$ | = | $2^{20} \cdot 3 \cdot 5^{2} \cdot 7^{7} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{12501706118329}{7156531200} \) | = | $2^{-20} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-1} \cdot 13^{-1} \cdot 23209^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5536090154755351810830238917$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.58065394094787852853034751998$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9797737929026421$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.24350649345013$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0991265818443107272533769092$ |
|
| Real period: | $\Omega$ | ≈ | $0.42882795502615907811371914053$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.3159684578380247675655884250 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.315968458 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.428828 \cdot 3.099127 \cdot 16}{2^2} \\ & \approx 5.315968458\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 122880 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{20}$ | nonsplit multiplicative | 1 | 1 | 20 | 20 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1460 & 1 \\ 751 & 6 \end{array}\right),\left(\begin{array}{rr} 1369 & 1368 \\ 286 & 1375 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 848 & 3 \\ 1349 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right),\left(\begin{array}{rr} 827 & 822 \\ 1370 & 275 \end{array}\right),\left(\begin{array}{rr} 1556 & 2183 \\ 1849 & 2178 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 6370 = 2 \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 19110.bl
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730.f4, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-91}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.427223104229376.7 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | split | split | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 2 | 2 | - | 1 | 2 | 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 | 0,0 |
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.