Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-2116x+345396\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-2116xz^2+345396z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2742363x+16123022838\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-321/4, 321/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 1690 \) | = | $2 \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-50983170062500$ | = | $-1 \cdot 2^{2} \cdot 5^{6} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{217081801}{10562500} \) | = | $-1 \cdot 2^{-2} \cdot 5^{-6} \cdot 13^{-2} \cdot 601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3101915271543497626275736077$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.027716848423581394600829886917$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9774623648888189$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.251146328784874$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.52477814212746819514895662584$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.0991125685098727805958265034 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.099112569 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.524778 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.099112569\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8064 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 4.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 13 & 24 \\ 852 & 253 \end{array}\right),\left(\begin{array}{rr} 1537 & 24 \\ 1536 & 25 \end{array}\right),\left(\begin{array}{rr} 1059 & 16 \\ 406 & 123 \end{array}\right),\left(\begin{array}{rr} 781 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 699 & 1550 \\ 148 & 1459 \end{array}\right),\left(\begin{array}{rr} 391 & 24 \\ 981 & 145 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 937 & 6 \\ 324 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$2415329280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 169 = 13^{2} \) |
| $3$ | good | $2$ | \( 338 = 2 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1690f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130a2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/6\Z\) | 2.2.13.1-1300.1-c1 |
| $4$ | 4.2.16900.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.160398576.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.182790400.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.4569760000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.285610000.1 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.20882706457600000000.2 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $18$ | 18.6.7333751368868611304482312500000000.2 | \(\Z/18\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | split | ord | nonsplit | add |
| $\lambda$-invariant(s) | 3 | 2 | 0 | - |
| $\mu$-invariant(s) | 1 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.