Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1167x+13741\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1167xz^2+13741z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18675x+860750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9, 58\right) \) | $0.53383929596396847986058296743$ | $\infty$ |
| \( \left(14, -7\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([9:58:1]\) | $0.53383929596396847986058296743$ | $\infty$ |
| \([14:-7:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(35, 500\right) \) | $0.53383929596396847986058296743$ | $\infty$ |
| \( \left(55, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-10, 161\right) \), \( \left(-10, -151\right) \), \( \left(9, 58\right) \), \( \left(9, -67\right) \), \( \left(14, -7\right) \), \( \left(39, 143\right) \), \( \left(39, -182\right) \), \( \left(134, 1433\right) \), \( \left(134, -1567\right) \)
\([-10:161:1]\), \([-10:-151:1]\), \([9:58:1]\), \([9:-67:1]\), \([14:-7:1]\), \([39:143:1]\), \([39:-182:1]\), \([134:1433:1]\), \([134:-1567:1]\)
\((-41,\pm 1248)\), \((35,\pm 500)\), \( \left(55, 0\right) \), \((155,\pm 1300)\), \((535,\pm 12000)\)
Invariants
| Conductor: | $N$ | = | \( 3150 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $23625000000$ | = | $2^{6} \cdot 3^{3} \cdot 5^{9} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( \frac{416832723}{56000} \) | = | $2^{-6} \cdot 3^{6} \cdot 5^{-3} \cdot 7^{-1} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.71792352502186942624613883060$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.36144850336220818390305214524$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9241940603148386$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.072007152192368$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.53383929596396847986058296743$ |
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| Real period: | $\Omega$ | ≈ | $1.1548396293792897534943044855$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4659950987965213264865450638 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.465995099 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.154840 \cdot 0.533839 \cdot 16}{2^2} \\ & \approx 2.465995099\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $1$ | $I_{1}$ | 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 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 421 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 675 & 808 \\ 638 & 797 \end{array}\right),\left(\begin{array}{rr} 152 & 11 \\ 525 & 808 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 829 & 12 \\ 828 & 13 \end{array}\right),\left(\begin{array}{rr} 730 & 3 \\ 213 & 832 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 790 & 831 \end{array}\right),\left(\begin{array}{rr} 830 & 837 \\ 699 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$743178240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 3150b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 630a3, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3 +4 \sqrt{-6}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.656373375.5 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.1372257936000000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4480842240000.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.91445760000.38 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.148365175523795581231125000000000000.4 | \(\Z/18\Z\) | 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 | add | add | nonsplit | ss | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | - | 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 |
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.