Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-38695x-3592802\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-38695xz^2-3592802z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-619115x-230558426\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{1227911624}{1129969}, \frac{41587462694622}{1201157047}\right) \) | $18.112724879058550287399884762$ | $\infty$ |
| \( \left(233, -117\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1305270056312:41587462694622:1201157047]\) | $18.112724879058550287399884762$ | $\infty$ |
| \([233:-117:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4910516527}{1129969}, \frac{337925586410412}{1201157047}\right) \) | $18.112724879058550287399884762$ | $\infty$ |
| \( \left(931, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(233, -117\right) \)
\([233:-117:1]\)
\( \left(931, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 17689 \) | = | $7^{2} \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1898470992842767$ | = | $-1 \cdot 7^{9} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -3375 \) | = | $-1 \cdot 3^{3} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6460361750552544654328783779$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2856159263194507434006498956$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9802957926219806$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.469763604940933$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $18.112724879058550287399884762$ |
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| Real period: | $\Omega$ | ≈ | $0.16763938541380179012307704401$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0364060668946527281325121914 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.036406067 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.167639 \cdot 18.112725 \cdot 4}{2^2} \\ & \approx 3.036406067\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 50400 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | additive | $20$ | \( 361 = 19^{2} \) |
| $19$ | additive | $182$ | \( 49 = 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 17689h
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49a1, its twist by $133$.
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 |
| $4$ | 4.2.1981168.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.495292.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.0.16468459.1 | \(\Z/14\Z\) | not in database |
| $8$ | 8.0.3925026644224.9 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3925026644224.10 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.33531379964523.2 | \(\Z/6\Z\) | not in database |
| $12$ | 12.0.13289296949899369.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $20$ | 20.0.68634231647302196434922446986223013.1 | \(\Z/2\Z \oplus \Z/22\Z\) | 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 | ss | ss | add | ord | ss | ss | add | ord | ord | ss | ord | ss | ord | ss |
| $\lambda$-invariant(s) | ? | 3,1 | 1,1 | - | 1 | 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 | 0 | 0,0 |
An entry ? indicates that the invariants have not yet been computed.
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.