Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-2927x+93943\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-2927xz^2+93943z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-46827x+5965542\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(47, 218\right) \) | $0.81913215084534044042396457841$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([47:218:1]\) | $0.81913215084534044042396457841$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(187, 1936\right) \) | $0.81913215084534044042396457841$ | $\infty$ |
Integral points
\( \left(-37, 406\right) \), \( \left(-37, -370\right) \), \( \left(47, 218\right) \), \( \left(47, -266\right) \)
\([-37:406:1]\), \([-37:-370:1]\), \([47:218:1]\), \([47:-266:1]\)
\((-149,\pm 3104)\), \((187,\pm 1936)\)
Invariants
| Conductor: | $N$ | = | \( 28314 \) | = | $2 \cdot 3^{2} \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2149002700416$ | = | $-1 \cdot 2^{7} \cdot 3^{6} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{2146689}{1664} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 13^{-1} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0650932323089680303719251057$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68316054842427208735666930174$ |
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| $abc$ quality: | $Q$ | ≈ | $0.96783604338842$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5516538616077162$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.81913215084534044042396457841$ |
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| Real period: | $\Omega$ | ≈ | $0.75667334735447087639075883770$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 14 $ = $ 7\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.6774165310813557052007065825 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.677416531 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.756673 \cdot 0.819132 \cdot 14}{1^2} \\ & \approx 8.677416531\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 39200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 | $24$ |
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 $96$, genus $2$, and generators
$\left(\begin{array}{rr} 16015 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12013 & 18942 \\ 21483 & 12475 \end{array}\right),\left(\begin{array}{rr} 19655 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 24011 & 14 \\ 24010 & 15 \end{array}\right),\left(\begin{array}{rr} 12937 & 18942 \\ 3927 & 12475 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 18019 & 6930 \\ 0 & 21451 \end{array}\right),\left(\begin{array}{rr} 6007 & 18942 \\ 3465 & 12475 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$535623421132800$ 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 | $4$ | \( 14157 = 3^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 3146 = 2 \cdot 11^{2} \cdot 13 \) |
| $7$ | good | $2$ | \( 14157 = 3^{2} \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 28314cb
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26b1, its twist by $33$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/7\Z\) | not in database |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.388694592.3 | \(\Z/14\Z\) | not in database |
| $8$ | 8.2.14632310742192.1 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/21\Z\) | 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 | split | add | ord | ord | add | split | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 1 | 1 | - | 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 |
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.