Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-12224x-48816\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-12224xz^2-48816z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-990171x-38557350\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-56, 676\right) \) | $1.2763351815336405780401699767$ | $\infty$ |
| \( \left(-8, 220\right) \) | $3.6791106910451880521274262787$ | $\infty$ |
| \( \left(-4, 0\right) \) | $0$ | $2$ |
| \( \left(113, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-56:676:1]\) | $1.2763351815336405780401699767$ | $\infty$ |
| \([-8:220:1]\) | $3.6791106910451880521274262787$ | $\infty$ |
| \([-4:0:1]\) | $0$ | $2$ |
| \([113:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-507, 18252\right) \) | $1.2763351815336405780401699767$ | $\infty$ |
| \( \left(-75, 5940\right) \) | $3.6791106910451880521274262787$ | $\infty$ |
| \( \left(-39, 0\right) \) | $0$ | $2$ |
| \( \left(1014, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-108, 0\right) \), \((-56,\pm 676)\), \((-40,\pm 612)\), \((-8,\pm 220)\), \( \left(-4, 0\right) \), \( \left(113, 0\right) \), \((117,\pm 330)\), \((230,\pm 3042)\), \((334,\pm 5746)\), \((725,\pm 19278)\), \((3038,\pm 167310)\)
\([-108:0:1]\), \([-56:\pm 676:1]\), \([-40:\pm 612:1]\), \([-8:\pm 220:1]\), \([-4:0:1]\), \([113:0:1]\), \([117:\pm 330:1]\), \([230:\pm 3042:1]\), \([334:\pm 5746:1]\), \([725:\pm 19278:1]\), \([3038:\pm 167310:1]\)
\( \left(-108, 0\right) \), \((-56,\pm 676)\), \((-40,\pm 612)\), \((-8,\pm 220)\), \( \left(-4, 0\right) \), \( \left(113, 0\right) \), \((117,\pm 330)\), \((230,\pm 3042)\), \((334,\pm 5746)\), \((725,\pm 19278)\), \((3038,\pm 167310)\)
Invariants
| Conductor: | $N$ | = | \( 137904 \) | = | $2^{4} \cdot 3 \cdot 13^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $115702550406144$ | = | $2^{10} \cdot 3^{4} \cdot 13^{6} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{40873252}{23409} \) | = | $2^{2} \cdot 3^{-4} \cdot 7^{3} \cdot 17^{-2} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3882056877125466759426647828$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.47189164148484278326510570587$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1382607398624434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.367086381507213$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.5291178790102522814838327432$ |
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| Real period: | $\Omega$ | ≈ | $0.49237701204139262215396686234$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.9201341138012704031193586727 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.920134114 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \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 0.492377 \cdot 4.529118 \cdot 64}{4^2} \\ & \approx 8.920134114\end{aligned}$$
Modular invariants
Modular form 137904.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 491520 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 8.12.0.4 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1768 = 2^{3} \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1765 & 4 \\ 1764 & 5 \end{array}\right),\left(\begin{array}{rr} 859 & 546 \\ 1014 & 1223 \end{array}\right),\left(\begin{array}{rr} 441 & 676 \\ 0 & 1767 \end{array}\right),\left(\begin{array}{rr} 885 & 546 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 271 & 0 \\ 0 & 1767 \end{array}\right)$.
The torsion field $K:=\Q(E[1768])$ is a degree-$65696956416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1768\Z)$.
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 | $2$ | \( 169 = 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 45968 = 2^{4} \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $86$ | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 137904cv
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 408b2, its twist by $-52$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-26}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.24439822686430441934952595456.3 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | ord | ord | ord | add | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | 2 | 2 | 4 | - | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.