Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+12607x-310943\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+12607xz^2-310943z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1021140x-223614000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(357, 7040\right) \) | $0.36490585507807008267316626364$ | $\infty$ |
| \( \left(\frac{397}{9}, \frac{15488}{27}\right) \) | $1.2639049641321801325411479938$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([357:7040:1]\) | $0.36490585507807008267316626364$ | $\infty$ |
| \([1191:15488:27]\) | $1.2639049641321801325411479938$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3210, 190080\right) \) | $0.36490585507807008267316626364$ | $\infty$ |
| \( \left(394, 15488\right) \) | $1.2639049641321801325411479938$ | $\infty$ |
Integral points
\((27,\pm 220)\), \((71,\pm 968)\), \((101,\pm 1408)\), \((192,\pm 3025)\), \((357,\pm 7040)\), \((632,\pm 16115)\), \((827,\pm 23980)\), \((3173,\pm 178816)\)
\([27:\pm 220:1]\), \([71:\pm 968:1]\), \([101:\pm 1408:1]\), \([192:\pm 3025:1]\), \([357:\pm 7040:1]\), \([632:\pm 16115:1]\), \([827:\pm 23980:1]\), \([3173:\pm 178816:1]\)
\((27,\pm 220)\), \((71,\pm 968)\), \((101,\pm 1408)\), \((192,\pm 3025)\), \((357,\pm 7040)\), \((632,\pm 16115)\), \((827,\pm 23980)\), \((3173,\pm 178816)\)
Invariants
| Conductor: | $N$ | = | \( 17600 \) | = | $2^{6} \cdot 5^{2} \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $-168874213376000$ | = | $-1 \cdot 2^{23} \cdot 5^{3} \cdot 11^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{6761990971}{5153632} \) | = | $2^{-5} \cdot 11^{-5} \cdot 31^{3} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4181298756247322179220855570$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.023950373323710839853952458494$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0570897476931809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.0856136534449154$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.44613533793638692943054845288$ |
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| Real period: | $\Omega$ | ≈ | $0.31976664676366705539867283196$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2^{2}\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.7063680405877547845367036428 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.706368041 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.319767 \cdot 0.446135 \cdot 40}{1^2} \\ & \approx 5.706368041\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46080 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{13}^{*}$ | additive | -1 | 6 | 23 | 5 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|---|
| $5$ | 5Cs.4.1 | 5.60.0.1 | $60$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 1099 & 2150 \\ 1075 & 949 \end{array}\right),\left(\begin{array}{rr} 1757 & 50 \\ 1351 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2151 & 50 \\ 2150 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 16 & 35 \\ 1765 & 1661 \end{array}\right),\left(\begin{array}{rr} 1649 & 2150 \\ 0 & 2199 \end{array}\right),\left(\begin{array}{rr} 11 & 50 \\ 645 & 1441 \end{array}\right)$.
The torsion field $K:=\Q(E[2200])$ is a degree-$5068800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2200\Z)$.
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 | $4$ | \( 55 = 5 \cdot 11 \) |
| $5$ | additive | $2$ | \( 64 = 2^{6} \) |
| $11$ | split multiplicative | $12$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 17600dd
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 550k2, its twist by $-8$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.440.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.8000.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.6195200.2 | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $8$ | 8.0.64000000.1 | \(\Z/5\Z \oplus \Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | 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 | ord | add | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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.