Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-593644x-132739125\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-593644xz^2-132739125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9498307x-8504802306\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{5838513321}{602176}, \frac{442395678465177}{467288576}\right) \) | $19.160950485524485733130912790$ | $\infty$ |
| \( \left(-614, 307\right) \) | $0$ | $2$ |
| \( \left(-250, 125\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4530686337096:442395678465177:467288576]\) | $19.160950485524485733130912790$ | $\infty$ |
| \([-614:307:1]\) | $0$ | $2$ |
| \([-250:125:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{5838362777}{150544}, \frac{444661021633725}{58411072}\right) \) | $19.160950485524485733130912790$ | $\infty$ |
| \( \left(-2457, 0\right) \) | $0$ | $2$ |
| \( \left(-1001, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-614, 307\right) \), \( \left(-250, 125\right) \)
\([-614:307:1]\), \([-250:125:1]\)
\( \left(-2457, 0\right) \), \( \left(-1001, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 41405 \) | = | $5 \cdot 7^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $5760593463285613225$ | = | $5^{2} \cdot 7^{10} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{40743095121}{10144225} \) | = | $3^{3} \cdot 5^{-2} \cdot 7^{-4} \cdot 13^{-2} \cdot 31^{3} \cdot 37^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3099848448869199331594646871$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.054555091628494912580044594598$ |
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| $abc$ quality: | $Q$ | ≈ | $1.041155076684769$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.843848098331742$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $19.160950485524485733130912790$ |
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| Real period: | $\Omega$ | ≈ | $0.17513392457055315943563022929$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7114649140638984436618038460 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.711464914 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.175134 \cdot 19.160950 \cdot 32}{4^2} \\ & \approx 6.711464914\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 516096 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $13$ | $4$ | $I_{2}^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 4.12.0.4 | $12$ |
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 $192$, genus $3$, and generators
$\left(\begin{array}{rr} 731 & 2 \\ 726 & 3639 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1825 & 4 \\ 3636 & 2727 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1825 & 4 \\ 3636 & 3637 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3119 & 3632 \\ 1556 & 3607 \end{array}\right),\left(\begin{array}{rr} 1957 & 3638 \\ 1122 & 1 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$202887659520$ 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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 845 = 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 245 = 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 41405.l
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 455.a2, its twist by $-91$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-35}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{35}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.10971993760000.7 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | add | ss | add | ord | ss | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 19,3 | 2 | - | 1,1 | - | 1 | 1,1 | 3 | 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 |
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.