Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-153x+357\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-153xz^2+357z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-198963x+19637262\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2, 7\right) \) | $1.0669826639952406382048183912$ | $\infty$ |
| \( \left(-14, 7\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2:7:1]\) | $1.0669826639952406382048183912$ | $\infty$ |
| \([-14:7:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(87, 1728\right) \) | $1.0669826639952406382048183912$ | $\infty$ |
| \( \left(-489, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-14, 7\right) \), \( \left(2, 7\right) \), \( \left(2, -9\right) \), \( \left(11, 7\right) \), \( \left(11, -18\right) \), \( \left(67, 511\right) \), \( \left(67, -578\right) \)
\([-14:7:1]\), \([2:7:1]\), \([2:-9:1]\), \([11:7:1]\), \([11:-18:1]\), \([67:511:1]\), \([67:-578:1]\)
\( \left(-489, 0\right) \), \((87,\pm 1728)\), \((411,\pm 2700)\), \((2427,\pm 117612)\)
Invariants
| Conductor: | $N$ | = | \( 9690 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $155040000$ | = | $2^{8} \cdot 3 \cdot 5^{4} \cdot 17 \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{400152624409}{155040000} \) | = | $2^{-8} \cdot 3^{-1} \cdot 5^{-4} \cdot 17^{-1} \cdot 19^{-1} \cdot 7369^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.27050554708620129123767631730$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.27050554708620129123767631730$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8649070250429451$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9105076051178522$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0669826639952406382048183912$ |
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| Real period: | $\Omega$ | ≈ | $1.6613101360996964608235699719$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7725891147379499233692034684 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.772589115 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.661310 \cdot 1.066983 \cdot 4}{2^2} \\ & \approx 1.772589115\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5120 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 38760 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 14539 & 14538 \\ 4858 & 24235 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 12928 & 3 \\ 12925 & 2 \end{array}\right),\left(\begin{array}{rr} 9124 & 1 \\ 13703 & 6 \end{array}\right),\left(\begin{array}{rr} 12244 & 1 \\ 12263 & 6 \end{array}\right),\left(\begin{array}{rr} 23257 & 8 \\ 15508 & 33 \end{array}\right),\left(\begin{array}{rr} 24233 & 24228 \\ 24230 & 4847 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 38754 & 38755 \end{array}\right),\left(\begin{array}{rr} 38753 & 8 \\ 38752 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[38760])$ is a degree-$7110865295769600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38760\Z)$.
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 | $4$ | \( 969 = 3 \cdot 17 \cdot 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3230 = 2 \cdot 5 \cdot 17 \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1938 = 2 \cdot 3 \cdot 17 \cdot 19 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 9690b
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{969}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-323}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-323})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.47534900625.1 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | nonsplit | ord | ord | ord | nonsplit | nonsplit | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 1 | 1 | 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 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.