Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-5139913x+4461601080\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-5139913xz^2+4461601080z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-82238611x+285460230510\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2616, 1308\right) \) | $0$ | $2$ |
| \( \left(1388, -694\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-2616:1308:1]\) | $0$ | $2$ |
| \([1388:-694:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-10465, 0\right) \) | $0$ | $2$ |
| \( \left(5551, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-2616, 1308\right) \), \( \left(1388, -694\right) \)
\([-2616:1308:1]\), \([1388:-694:1]\)
\( \left(-10465, 0\right) \), \( \left(5551, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 91091 \) | = | $7^{2} \cdot 11 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $96161939372022452929$ | = | $7^{8} \cdot 11^{2} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{26444947540257}{169338169} \) | = | $3^{3} \cdot 7^{-2} \cdot 11^{-2} \cdot 13^{-4} \cdot 9931^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6710840811617346753109379351$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41565432790330965473151784260$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0317832975087151$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.076462464037728$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.19083485074570328673408990793$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.38166970149140657346817981587 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.381669701 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.190835 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 0.381669701\end{aligned}$$
Modular invariants
Modular form 91091.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2451456 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 571 & 1846 \\ 0 & 4003 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 365 & 312 \\ 2886 & 625 \end{array}\right),\left(\begin{array}{rr} 157 & 312 \\ 2158 & 2159 \end{array}\right),\left(\begin{array}{rr} 4001 & 4 \\ 4000 & 5 \end{array}\right),\left(\begin{array}{rr} 3079 & 0 \\ 0 & 4003 \end{array}\right)$.
The torsion field $K:=\Q(E[4004])$ is a degree-$1394852659200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4004\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 1859 = 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 539 = 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 91091b
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1001b2, its twist by $-91$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{77})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{-91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 7 | 11 | 13 |
|---|---|---|---|---|
| Reduction type | ord | add | nonsplit | add |
| $\lambda$-invariant(s) | 5 | - | 0 | - |
| $\mu$-invariant(s) | 0 | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.