Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1514102x+528518310\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1514102xz^2+528518310z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24225627x+33800946230\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-223, 29352\right) \) | $2.6675208810713809890773555857$ | $\infty$ |
| \( \left(-\frac{5509}{4}, \frac{5505}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-223:29352:1]\) | $2.6675208810713809890773555857$ | $\infty$ |
| \([-11018:5505:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-893, 233928\right) \) | $2.6675208810713809890773555857$ | $\infty$ |
| \( \left(-5510, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-223, 29352\right) \), \( \left(-223, -29130\right) \)
\([-223:29352:1]\), \([-223:-29130:1]\)
\((-893,\pm 233928)\)
Invariants
| Conductor: | $N$ | = | \( 22743 \) | = | $3^{2} \cdot 7 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $101650914287951576247$ | = | $3^{8} \cdot 7 \cdot 19^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{11192824869409}{2963890503} \) | = | $3^{-2} \cdot 7^{-1} \cdot 19^{-6} \cdot 22369^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5475665526669506147203973525$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52604091874967553901826101809$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9564366131642995$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.413131279426151$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6675208810713809890773555857$ |
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| Real period: | $\Omega$ | ≈ | $0.17657583586762689839613871312$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.8840789170781106448014297413 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.884078917 \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 0.176576 \cdot 2.667521 \cdot 16}{2^2} \\ & \approx 1.884078917\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 967680 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 400 & 1201 \\ 1197 & 400 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 533 & 4 \\ 1066 & 9 \end{array}\right),\left(\begin{array}{rr} 1593 & 4 \\ 1592 & 5 \end{array}\right),\left(\begin{array}{rr} 230 & 1 \\ 683 & 0 \end{array}\right),\left(\begin{array}{rr} 1009 & 4 \\ 422 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[1596])$ is a degree-$95312609280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1596\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 63 = 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 22743o
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 399c2, its twist by $57$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.90972.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6488309350656.18 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | ord | nonsplit | ord | ord | ss | add | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | - | 1 | 1 | 1 | 1 | 1,1 | - | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.