Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+504x+4464\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+504xz^2+4464z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+652509x+198481374\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 82)$ | $0.83615559653618216807084424650$ | $\infty$ |
| $(-8, 4)$ | $0$ | $2$ |
Integral points
\( \left(-8, 4\right) \), \( \left(1, 70\right) \), \( \left(1, -71\right) \), \( \left(5, 82\right) \), \( \left(5, -87\right) \), \( \left(44, 316\right) \), \( \left(44, -360\right) \)
Invariants
| Conductor: | $N$ | = | \( 23322 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-15986391408$ | = | $-1 \cdot 2^{4} \cdot 3^{2} \cdot 13^{6} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{2924207}{3312} \) | = | $2^{-4} \cdot 3^{-2} \cdot 11^{3} \cdot 13^{3} \cdot 23^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.64434175154801290860951260525$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63813292718275545941723111553$ |
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| $abc$ quality: | $Q$ | ≈ | $0.898775602359137$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.010616607073316$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.83615559653618216807084424650$ |
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| Real period: | $\Omega$ | ≈ | $0.82524454916874179230661680646$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7601313931936880379127591634 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.760131393 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.825245 \cdot 0.836156 \cdot 16}{2^2} \\ & \approx 2.760131393\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18432 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | nonsplit 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 | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2392 = 2^{3} \cdot 13 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 859 & 1222 \\ 442 & 2003 \end{array}\right),\left(\begin{array}{rr} 2385 & 8 \\ 2384 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2386 & 2387 \end{array}\right),\left(\begin{array}{rr} 625 & 624 \\ 910 & 79 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 2391 \end{array}\right),\left(\begin{array}{rr} 40 & 923 \\ 117 & 1106 \end{array}\right)$.
The torsion field $K:=\Q(E[2392])$ is a degree-$224062046208$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2392\Z)$.
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 | $4$ | \( 3887 = 13^{2} \cdot 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 7774 = 2 \cdot 13^{2} \cdot 23 \) |
| $13$ | additive | $86$ | \( 138 = 2 \cdot 3 \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 23322b
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 138c1, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-299}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.5012726943744.50 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.17318105193385984.6 | \(\Z/8\Z\) | 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 | nonsplit | ord | ss | ss | add | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 3 | 1 | 1,1 | 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 | 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.