Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+4278x+15614\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+4278xz^2+15614z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+68445x+1067742\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{25}{9}, \frac{4441}{27}\right) \) | $3.8591653925831260945758401293$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([75:4441:27]\) | $3.8591653925831260945758401293$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{91}{9}, \frac{35828}{27}\right) \) | $3.8591653925831260945758401293$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 27378 \) | = | $2 \cdot 3^{4} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-5130328403538$ | = | $-1 \cdot 2 \cdot 3^{12} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3375}{2} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1281938274148067430025302182$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2528931399840713164194587395$ |
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| $abc$ quality: | $Q$ | ≈ | $1.4265653296335434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5916038159993846$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.8591653925831260945758401293$ |
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| Real period: | $\Omega$ | ≈ | $0.46704900549047514533265386624$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.6048387172584162581022780271 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.604838717 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.467049 \cdot 3.859165 \cdot 2}{1^2} \\ & \approx 3.604838717\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41472 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 4 | 12 | 0 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $7$ | 7B | 7.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 4537 & 2016 \\ 4536 & 4537 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5460 & 1 \end{array}\right),\left(\begin{array}{rr} 1093 & 546 \\ 5733 & 547 \end{array}\right),\left(\begin{array}{rr} 4369 & 2184 \\ 4368 & 2185 \end{array}\right),\left(\begin{array}{rr} 1 & 546 \\ 0 & 5617 \end{array}\right),\left(\begin{array}{rr} 5511 & 2366 \\ 5096 & 2391 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6006 & 1 \end{array}\right),\left(\begin{array}{rr} 3823 & 546 \\ 6279 & 2731 \end{array}\right),\left(\begin{array}{rr} 1 & 2184 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6474 \\ 546 & 3277 \end{array}\right),\left(\begin{array}{rr} 6007 & 1794 \\ 5460 & 3823 \end{array}\right),\left(\begin{array}{rr} 3550 & 1833 \\ 5733 & 2185 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2016 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3744 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5825 & 2912 \\ 3640 & 2913 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2184 & 1 \end{array}\right),\left(\begin{array}{rr} 4535 & 0 \\ 0 & 6551 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$410847510528$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
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 | $4$ | \( 13689 = 3^{4} \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 27378b
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c1, its twist by $-39$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.25625808.3 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.14414517.1 | \(\Z/7\Z\) | not in database |
| $6$ | 6.0.2767587264.5 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.5910138320875776.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.1870004703089601.1 | \(\Z/21\Z\) | not in database |
| $18$ | 18.0.891136206059071970130118099934976.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.146523640191573076809272449302528.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.785127530176038863218409472.1 | \(\Z/14\Z\) | not in database |
| $18$ | 18.6.8943093273411442676347195392.3 | \(\Z/21\Z\) | 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 | add | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1,1 | 1 | 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 |
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.