Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1230827x+568122906\)
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(homogenize, simplify) |
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\(y^2z=x^3-1230827xz^2+568122906z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1230827x+568122906\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(455, 10114)$ | $4.3158514236390882841207786909$ | $\infty$ |
Integral points
\((455,\pm 10114)\)
Invariants
| Conductor: | $N$ | = | \( 13520 \) | = | $2^{4} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-20098017024582615040$ | = | $-1 \cdot 2^{47} \cdot 5 \cdot 13^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{1762712152495281}{171798691840} \) | = | $-1 \cdot 2^{-35} \cdot 3^{3} \cdot 5^{-1} \cdot 13^{2} \cdot 7283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4449327322441051112723243133$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89680243253031422317059637799$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1377369369986685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.660141373381435$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3158514236390882841207786909$ |
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| Real period: | $\Omega$ | ≈ | $0.21108830617249119806926558056$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4661546002486593041917221129 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.466154600 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.211088 \cdot 4.315851 \cdot 6}{1^2} \\ & \approx 5.466154600\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 282240 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{39}^{*}$ | additive | -1 | 4 | 47 | 35 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1457 & 14 \\ 2919 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 3627 & 14 \\ 3626 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2531 & 2 \\ 1750 & 3551 \end{array}\right),\left(\begin{array}{rr} 2729 & 3626 \\ 903 & 3541 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 1821 & 14 \\ 1827 & 99 \end{array}\right),\left(\begin{array}{rr} 913 & 528 \\ 1806 & 3473 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$405775319040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 845 = 5 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 2704 = 2^{4} \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 80 = 2^{4} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 13520w
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 1690g2, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.6760.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.30721582528.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.1855712631115861770653408771964928000000.1 | \(\Z/14\Z\) | 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 | add | ss | split | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1,1 | 2 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 3 |
| $\mu$-invariant(s) | - | 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.