Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-7665x+2234668\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-7665xz^2+2234668z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9933219x+104290481502\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-112, 1356\right) \) | $0.97223249795186481427180137796$ | $\infty$ |
| \( \left(-\frac{601}{4}, \frac{597}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-112:1356:1]\) | $0.97223249795186481427180137796$ | $\infty$ |
| \([-1202:597:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-4029, 280908\right) \) | $0.97223249795186481427180137796$ | $\infty$ |
| \( \left(-5406, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-112, 1356\right) \), \( \left(-112, -1245\right) \), \( \left(356, 6504\right) \), \( \left(356, -6861\right) \), \( \left(1418, 52611\right) \), \( \left(1418, -54030\right) \)
\([-112:1356:1]\), \([-112:-1245:1]\), \([356:6504:1]\), \([356:-6861:1]\), \([1418:52611:1]\), \([1418:-54030:1]\)
\((-4029,\pm 280908)\), \((12819,\pm 1443420)\), \((51051,\pm 11517228)\)
Invariants
| Conductor: | $N$ | = | \( 71094 \) | = | $2 \cdot 3 \cdot 17^{2} \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2129713905130632$ | = | $-1 \cdot 2^{3} \cdot 3^{8} \cdot 17^{6} \cdot 41^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{2062933417}{88232328} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-8} \cdot 19^{3} \cdot 41^{-2} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6213163211994312584262289472$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.20470964917132321830146163826$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0089033593136165$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8277702504227764$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.97223249795186481427180137796$ |
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| Real period: | $\Omega$ | ≈ | $0.38520863510353735422291080351$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 1\cdot2^{3}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9921976566294483414021881176 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.992197657 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.385209 \cdot 0.972232 \cdot 64}{2^2} \\ & \approx 5.992197657\end{aligned}$$
Modular invariants
Modular form 71094.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 491520 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $17$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $41$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 8.6.0.5 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 328 = 2^{3} \cdot 41 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 325 & 4 \\ 324 & 5 \end{array}\right),\left(\begin{array}{rr} 124 & 209 \\ 41 & 288 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 129 & 4 \\ 258 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 163 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[328])$ is a degree-$352665600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/328\Z)$.
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 | $4$ | \( 289 = 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 11849 = 17^{2} \cdot 41 \) |
| $17$ | additive | $146$ | \( 246 = 2 \cdot 3 \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 71094n
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 246d2, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{102 +34 \sqrt{41}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.15467176557346816.29 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | \(\Z/8\Z\) | 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 | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | split | ord | ord |
| $\lambda$-invariant(s) | 11 | 2 | 1 | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 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.