Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-307040x+65327681\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-307040xz^2+65327681z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-397923867x+3053897151030\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(347, 673\right) \) | $0.73297195571129050530561532860$ | $\infty$ |
| \( \left(325, -163\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([347:673:1]\) | $0.73297195571129050530561532860$ | $\infty$ |
| \([325:-163:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(12507, 182952\right) \) | $0.73297195571129050530561532860$ | $\infty$ |
| \( \left(11715, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(325, -163\right) \), \( \left(347, 673\right) \), \( \left(347, -1021\right) \), \( \left(361, 1121\right) \), \( \left(361, -1483\right) \), \( \left(809, 18229\right) \), \( \left(809, -19039\right) \), \( \left(22853, 3442397\right) \), \( \left(22853, -3465251\right) \)
\([325:-163:1]\), \([347:673:1]\), \([347:-1021:1]\), \([361:1121:1]\), \([361:-1483:1]\), \([809:18229:1]\), \([809:-19039:1]\), \([22853:3442397:1]\), \([22853:-3465251:1]\)
\( \left(11715, 0\right) \), \((12507,\pm 182952)\), \((13011,\pm 281232)\), \((29139,\pm 4024944)\), \((822723,\pm 746025984)\)
Invariants
| Conductor: | $N$ | = | \( 66066 \) | = | $2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $1651667760654912$ | = | $2^{6} \cdot 3^{3} \cdot 7^{3} \cdot 11^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{1806976738085401}{932323392} \) | = | $2^{-6} \cdot 3^{-3} \cdot 7^{-3} \cdot 11^{-2} \cdot 13^{-1} \cdot 349^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8712384359137787784979847543$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.67229079951459350646701296532$ |
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| $abc$ quality: | $Q$ | ≈ | $1.09371239803995$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.461702649806236$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.73297195571129050530561532860$ |
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| Real period: | $\Omega$ | ≈ | $0.46731387972974937757998100173$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ ( 2 \cdot 3 )\cdot1\cdot3\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.1655034304178135746943908845 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.165503430 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.467314 \cdot 0.732972 \cdot 72}{2^2} \\ & \approx 6.165503430\end{aligned}$$
Modular invariants
Modular form 66066.2.a.bl
For more coefficients, see the Downloads section to the right.
| Modular degree: | 829440 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $1$ | $I_{1}$ | split 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 21025 & 3004 \\ 9008 & 15015 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 4369 & 4 \\ 8738 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 24021 & 4 \\ 24020 & 5 \end{array}\right),\left(\begin{array}{rr} 12013 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1850 & 1 \\ 11087 & 0 \end{array}\right),\left(\begin{array}{rr} 6866 & 1 \\ 20591 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 16018 & 1 \\ 16015 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$4284987369062400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
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 | $4$ | \( 33033 = 3 \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1573 = 11^{2} \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $72$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 66066.bl
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 6006.a2, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.2114112.2 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | nonsplit | ord | split | add | split | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 5 | 1 | 2 | - | 2 | 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 |
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.