Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-193x+1032\)
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(homogenize, simplify) |
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\(y^2z=x^3-193xz^2+1032z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-193x+1032\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(4, 18\right) \) | $1.9128913154739133011142675675$ | $\infty$ |
| \( \left(8, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4:18:1]\) | $1.9128913154739133011142675675$ | $\infty$ |
| \([8:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(4, 18\right) \) | $1.9128913154739133011142675675$ | $\infty$ |
| \( \left(8, 0\right) \) | $0$ | $2$ |
Integral points
\((4,\pm 18)\), \( \left(8, 0\right) \)
\([4:\pm 18:1]\), \([8:0:1]\)
\((4,\pm 18)\), \( \left(8, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 4640 \) | = | $2^{5} \cdot 5 \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $9280$ | = | $2^{6} \cdot 5 \cdot 29 $ |
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| j-invariant: | $j$ | = | \( \frac{12422690496}{145} \) | = | $2^{6} \cdot 3^{3} \cdot 5^{-1} \cdot 29^{-1} \cdot 193^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.087988454087092309848160400759$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.43456204436706496455677646149$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9997268968596169$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.245694062394618$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9128913154739133011142675675$ |
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| Real period: | $\Omega$ | ≈ | $3.7237049313394668152698424601$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5615214122733253420234773990 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.561521412 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.723705 \cdot 1.912891 \cdot 2}{2^2} \\ & \approx 3.561521412\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 384 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 5 | 6 | 0 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $1$ | $I_{1}$ | 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 | 8.12.0.22 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1160 = 2^{3} \cdot 5 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 248 & 3 \\ 445 & 2 \end{array}\right),\left(\begin{array}{rr} 289 & 1152 \\ 580 & 1159 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 936 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 1153 & 8 \\ 1152 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 587 & 6 \\ 1154 & 1155 \end{array}\right)$.
The torsion field $K:=\Q(E[1160])$ is a degree-$10476748800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1160\Z)$.
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 | $2$ | \( 145 = 5 \cdot 29 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 928 = 2^{5} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 160 = 2^{5} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 4640.d
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4640.c1, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2320.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.113164960000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | ss | nonsplit | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | 3,5 | 1 | 1 | 1 | 3 | 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 | 0,0 | 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.