Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-2013x-12344\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-2013xz^2-12344z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2608227x-568085346\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(95, 762\right) \) | $0.092268978551126786228297544757$ | $\infty$ |
| \( \left(-\frac{25}{4}, \frac{21}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([95:762:1]\) | $0.092268978551126786228297544757$ | $\infty$ |
| \([-50:21:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3423, 174960\right) \) | $0.092268978551126786228297544757$ | $\infty$ |
| \( \left(-222, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-40, 87\right) \), \( \left(-40, -48\right) \), \( \left(-25, 162\right) \), \( \left(-25, -138\right) \), \( \left(-13, 114\right) \), \( \left(-13, -102\right) \), \( \left(-10, 87\right) \), \( \left(-10, -78\right) \), \( \left(50, 87\right) \), \( \left(50, -138\right) \), \( \left(55, 182\right) \), \( \left(55, -238\right) \), \( \left(95, 762\right) \), \( \left(95, -858\right) \), \( \left(275, 4362\right) \), \( \left(275, -4638\right) \), \( \left(500, 10887\right) \), \( \left(500, -11388\right) \), \( \left(4055, 256182\right) \), \( \left(4055, -260238\right) \)
\([-40:87:1]\), \([-40:-48:1]\), \([-25:162:1]\), \([-25:-138:1]\), \([-13:114:1]\), \([-13:-102:1]\), \([-10:87:1]\), \([-10:-78:1]\), \([50:87:1]\), \([50:-138:1]\), \([55:182:1]\), \([55:-238:1]\), \([95:762:1]\), \([95:-858:1]\), \([275:4362:1]\), \([275:-4638:1]\), \([500:10887:1]\), \([500:-11388:1]\), \([4055:256182:1]\), \([4055:-260238:1]\)
\((-1437,\pm 14580)\), \((-897,\pm 32400)\), \((-465,\pm 23328)\), \((-357,\pm 17820)\), \((1803,\pm 24300)\), \((1983,\pm 45360)\), \((3423,\pm 174960)\), \((9903,\pm 972000)\), \((18003,\pm 2405700)\), \((145983,\pm 55773360)\)
Invariants
| Conductor: | $N$ | = | \( 930 \) | = | $2 \cdot 3 \cdot 5 \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $457629750000$ | = | $2^{4} \cdot 3^{10} \cdot 5^{6} \cdot 31 $ |
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| j-invariant: | $j$ | = | \( \frac{901456690969801}{457629750000} \) | = | $2^{-4} \cdot 3^{-10} \cdot 5^{-6} \cdot 31^{-1} \cdot 96601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.92990978483353069901192563807$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.92990978483353069901192563807$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9986284563216552$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.03790829381994$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.092268978551126786228297544757$ |
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| Real period: | $\Omega$ | ≈ | $0.75218734996805946773255636564$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 120 $ = $ 2\cdot( 2 \cdot 5 )\cdot( 2 \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0821067538189532968114988646 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.082106754 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.752187 \cdot 0.092269 \cdot 120}{2^2} \\ & \approx 2.082106754\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1920 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $31$ | $1$ | $I_{1}$ | 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 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1857 & 4 \\ 1856 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1396 & 469 \\ 465 & 1396 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1117 & 4 \\ 374 & 9 \end{array}\right),\left(\begin{array}{rr} 1241 & 4 \\ 622 & 9 \end{array}\right),\left(\begin{array}{rr} 1742 & 1 \\ 59 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1860])$ is a degree-$164560896000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1860\Z)$.
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 | $4$ | \( 31 \) |
| $3$ | split multiplicative | $4$ | \( 62 = 2 \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 62 = 2 \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 930h
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{31}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{4 + \sqrt{-15}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.11968832160000.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.2617583593392.6 | \(\Z/6\Z\) | 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 | nonsplit | split | split | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | 4 | 2 | 1 | 1 | 1 | 1 | 3 | 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 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.