Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+430484x+453494672\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+430484xz^2+453494672z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+557907237x+21156573695094\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(176, 23036\right) \) | $4.4199104274240500580387096773$ | $\infty$ |
| \( \left(344, 25172\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([176:23036:1]\) | $4.4199104274240500580387096773$ | $\infty$ |
| \([344:25172:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6339, 4994784\right) \) | $4.4199104274240500580387096773$ | $\infty$ |
| \( \left(12387, 5474304\right) \) | $0$ | $7$ |
Integral points
\( \left(-448, 13292\right) \), \( \left(-448, -12844\right) \), \( \left(176, 23036\right) \), \( \left(176, -23212\right) \), \( \left(344, 25172\right) \), \( \left(344, -25516\right) \), \( \left(2456, 126548\right) \), \( \left(2456, -129004\right) \), \( \left(82712, 23747156\right) \), \( \left(82712, -23829868\right) \)
\([-448:13292:1]\), \([-448:-12844:1]\), \([176:23036:1]\), \([176:-23212:1]\), \([344:25172:1]\), \([344:-25516:1]\), \([2456:126548:1]\), \([2456:-129004:1]\), \([82712:23747156:1]\), \([82712:-23829868:1]\)
\((-16125,\pm 2822688)\), \((6339,\pm 4994784)\), \((12387,\pm 5474304)\), \((88419,\pm 27599616)\), \((2977635,\pm 5138318592)\)
Invariants
| Conductor: | $N$ | = | \( 69366 \) | = | $2 \cdot 3 \cdot 11 \cdot 1051$ |
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| Minimal Discriminant: | $\Delta$ | = | $-93935597925332680704$ | = | $-1 \cdot 2^{21} \cdot 3^{7} \cdot 11^{7} \cdot 1051 $ |
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| j-invariant: | $j$ | = | \( \frac{8822561460536124355391}{93935597925332680704} \) | = | $2^{-21} \cdot 3^{-7} \cdot 11^{-7} \cdot 97^{3} \cdot 1051^{-1} \cdot 213023^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5134696460212187668208989430$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5134696460212187668208989430$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9773037383757964$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.789382520163407$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.4199104274240500580387096773$ |
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| Real period: | $\Omega$ | ≈ | $0.13998118069041130253231448728$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1029 $ = $ ( 3 \cdot 7 )\cdot7\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.992789883710259058323810846 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.992789884 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.139981 \cdot 4.419910 \cdot 1029}{7^2} \\ & \approx 12.992789884\end{aligned}$$
Modular invariants
Modular form 69366.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1893360 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $11$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $1051$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1942248 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 1051 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1009016 & 7 \\ 1544921 & 1942242 \end{array}\right),\left(\begin{array}{rr} 1589120 & 7 \\ 1765673 & 1942242 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 485563 & 971138 \\ 0 & 208099 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 971117 & 1942242 \end{array}\right),\left(\begin{array}{rr} 1456687 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1942235 & 14 \\ 1942234 & 15 \end{array}\right),\left(\begin{array}{rr} 1294840 & 7 \\ 1294825 & 1942242 \end{array}\right)$.
The torsion field $K:=\Q(E[1942248])$ is a degree-$24912810984112128000000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1942248\Z)$.
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 | $4$ | \( 34683 = 3 \cdot 11 \cdot 1051 \) |
| $3$ | split multiplicative | $4$ | \( 11561 = 11 \cdot 1051 \) |
| $7$ | good | $2$ | \( 1051 \) |
| $11$ | split multiplicative | $12$ | \( 6306 = 2 \cdot 3 \cdot 1051 \) |
| $1051$ | split multiplicative | $1052$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 69366w
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.277464.4 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.21360918778873344.3 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | 1051 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | ord | ord | split | ss | ord | ord | ord | ord | ord | ord | ss | ord | ord | split |
| $\lambda$-invariant(s) | 3 | 4 | 1 | 3 | 4 | 1,1 | 5 | 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 | ? |
An entry ? indicates that the invariants have not yet been computed.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.