Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+227580x-294201648\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+227580xz^2-294201648z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+294943005x-13730696237538\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4424, 293292)$ | $4.6622169052748955800219616890$ | $\infty$ |
| $(552, -276)$ | $0$ | $2$ |
Integral points
\( \left(552, -276\right) \), \( \left(4424, 293292\right) \), \( \left(4424, -297716\right) \)
Invariants
| Conductor: | $N$ | = | \( 6762 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-38170059896287484928$ | = | $-1 \cdot 2^{10} \cdot 3^{12} \cdot 7^{8} \cdot 23^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{11079872671250375}{324440155855872} \) | = | $2^{-10} \cdot 3^{-12} \cdot 5^{3} \cdot 7^{-2} \cdot 23^{-3} \cdot 44587^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4374372388669841497099822243$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4644821643393274971573058526$ |
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| $abc$ quality: | $Q$ | ≈ | $1.031710352925848$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.955652724288321$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6622169052748955800219616890$ |
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| Real period: | $\Omega$ | ≈ | $0.098967053083637135273508700615$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7684352077106262185104885907 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.768435208 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098967 \cdot 4.662217 \cdot 24}{2^2} \\ & \approx 2.768435208\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 230400 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $7$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $23$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 8.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 3137 & 1106 \\ 1806 & 2773 \end{array}\right),\left(\begin{array}{rr} 1275 & 2464 \\ 686 & 3557 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2626 & 1659 \\ 693 & 2752 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3814 & 3855 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 1933 & 2772 \\ 3318 & 1177 \end{array}\right),\left(\begin{array}{rr} 3853 & 12 \\ 3852 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$413653008384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1127 = 7^{2} \cdot 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 98 = 2 \cdot 7^{2} \) |
| $5$ | good | $2$ | \( 3381 = 3 \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $32$ | \( 138 = 2 \cdot 3 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 6762f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 966f1, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.72128.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.7260624.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2752095195136.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.363964589556736.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5202448384.6 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.52716660869376.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.111914448601773322205359052009500439746023168.1 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | ss | add | ord | ord | ord | ord | split | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 1 | 1,1 | - | 1 | 1 | 1 | 1 | 2 | 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 |
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.