Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1192251x-450858134\)
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(homogenize, simplify) |
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\(y^2z=x^3-1192251xz^2-450858134z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1192251x-450858134\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-481, 3366)$ | $1.1074821963495119359741788203$ | $\infty$ |
| $(-787, 0)$ | $0$ | $2$ |
Integral points
\( \left(-787, 0\right) \), \((-481,\pm 3366)\), \((1389,\pm 23936)\), \((6438,\pm 508640)\)
Invariants
| Conductor: | $N$ | = | \( 26928 \) | = | $2^{4} \cdot 3^{2} \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $20649400908169347072$ | = | $2^{20} \cdot 3^{11} \cdot 11^{3} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{62768149033310713}{6915442583808} \) | = | $2^{-8} \cdot 3^{-5} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 37^{3} \cdot 467^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4395721334747446656036657746$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1971188085807445104888110347$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9879715979533532$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.253217748009752$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1074821963495119359741788203$ |
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| Real period: | $\Omega$ | ≈ | $0.14551332122554620729394098470$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot2\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8676819021355192215585168285 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.867681902 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.145513 \cdot 1.107482 \cdot 96}{2^2} \\ & \approx 3.867681902\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{12}^{*}$ | additive | -1 | 4 | 20 | 8 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 4482 & 4483 \end{array}\right),\left(\begin{array}{rr} 1492 & 4487 \\ 2969 & 4482 \end{array}\right),\left(\begin{array}{rr} 3272 & 3 \\ 2045 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1679 & 1680 \\ 3914 & 1673 \end{array}\right),\left(\begin{array}{rr} 1057 & 8 \\ 4228 & 33 \end{array}\right),\left(\begin{array}{rr} 4481 & 8 \\ 4480 & 9 \end{array}\right),\left(\begin{array}{rr} 3931 & 3930 \\ 2818 & 571 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$1588278067200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 | $2$ | \( 99 = 3^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 272 = 2^{4} \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 26928.m
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c3, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.143748.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.330615800064.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1886029795584.9 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3068151359682816.25 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.15150586457088.12 | \(\Z/6\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | add | ord | ord | split | ord | split | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 2 | 1 | 2 | 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 |
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.