Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-48x+297\)
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(homogenize, simplify) |
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\(y^2z=x^3-48xz^2+297z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-48x+297\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 15\right) \) | $0.53622331782395737612214814424$ | $\infty$ |
| \( \left(-9, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([6:15:1]\) | $0.53622331782395737612214814424$ | $\infty$ |
| \([-9:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 15\right) \) | $0.53622331782395737612214814424$ | $\infty$ |
| \( \left(-9, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-9, 0\right) \), \((-8,\pm 13)\), \((4,\pm 13)\), \((6,\pm 15)\), \((186,\pm 2535)\)
\([-9:0:1]\), \([-8:\pm 13:1]\), \([4:\pm 13:1]\), \([6:\pm 15:1]\), \([186:\pm 2535:1]\)
\( \left(-9, 0\right) \), \((-8,\pm 13)\), \((4,\pm 13)\), \((6,\pm 15)\), \((186,\pm 2535)\)
Invariants
| Conductor: | $N$ | = | \( 39780 \) | = | $2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $-31028400$ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 13^{2} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( -\frac{28311552}{71825} \) | = | $-1 \cdot 2^{20} \cdot 3^{3} \cdot 5^{-2} \cdot 13^{-2} \cdot 17^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.12532965762889238511652620941$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.38037247472478347420469580697$ |
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| $abc$ quality: | $Q$ | ≈ | $1.016987102247835$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.3520280072327115$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.53622331782395737612214814424$ |
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| Real period: | $\Omega$ | ≈ | $1.8437872022412671799419092146$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9320901456829851207879846159 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.932090146 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.843787 \cdot 0.536223 \cdot 24}{2^2} \\ & \approx 5.932090146\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7680 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $1$ | $I_{1}$ | nonsplit 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 \( 13260 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4424 & 1 \\ 8839 & 0 \end{array}\right),\left(\begin{array}{rr} 9949 & 3316 \\ 9944 & 3315 \end{array}\right),\left(\begin{array}{rr} 7957 & 4 \\ 2654 & 9 \end{array}\right),\left(\begin{array}{rr} 11702 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 13257 & 4 \\ 13256 & 5 \end{array}\right),\left(\begin{array}{rr} 11221 & 4 \\ 9182 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[13260])$ is a degree-$378414468956160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13260\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 51 = 3 \cdot 17 \) |
| $3$ | additive | $6$ | \( 4420 = 2^{2} \cdot 5 \cdot 13 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7956 = 2^{2} \cdot 3^{2} \cdot 13 \cdot 17 \) |
| $13$ | split multiplicative | $14$ | \( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 39780.i
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-51}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-27 +2 \sqrt{195}})\) | \(\Z/4\Z\) | 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 | add | add | nonsplit | ord | ss | split | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1,3 | 2 | 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.