Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-4379x-22822\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-4379xz^2-22822z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-70059x-1530650\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-63, 31)$ | $0$ | $2$ |
| $(69, -35)$ | $0$ | $2$ |
Integral points
\( \left(-63, 31\right) \), \( \left(69, -35\right) \)
Invariants
| Conductor: | $N$ | = | \( 7623 \) | = | $3^{2} \cdot 7 \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $5125836368961$ | = | $3^{10} \cdot 7^{2} \cdot 11^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{7189057}{3969} \) | = | $3^{-4} \cdot 7^{-2} \cdot 193^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1293119835642345790696281407$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.61894179716900553865896626674$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1486166056524132$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.11314725392435$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.62823289868426462977666365930$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.2564657973685292595533273186 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.256465797 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.628233 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 1.256465797\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10240 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1844 & 1845 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1841 & 8 \\ 1840 & 9 \end{array}\right),\left(\begin{array}{rr} 1519 & 1518 \\ 66 & 331 \end{array}\right),\left(\begin{array}{rr} 89 & 1056 \\ 1716 & 749 \end{array}\right),\left(\begin{array}{rr} 1231 & 836 \\ 0 & 1847 \end{array}\right),\left(\begin{array}{rr} 1013 & 132 \\ 330 & 1211 \end{array}\right),\left(\begin{array}{rr} 1175 & 0 \\ 0 & 1847 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$10218700800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
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 | $2$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 847 = 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 63 = 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 7623p
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21a1, its twist by $33$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.33.1-147.1-a3 |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.728933458176.8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.728933458176.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.6227255754027.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
| $16$ | 16.0.531343986448422341246976.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 11 |
|---|---|---|---|---|
| Reduction type | ord | add | split | add |
| $\lambda$-invariant(s) | 4 | - | 1 | - |
| $\mu$-invariant(s) | 0 | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 $0$.