Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-264x+6624\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-264xz^2+6624z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-21411x+4764690\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(14, 74\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([14:74:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(123, 1998\right) \) | $0$ | $4$ |
Integral points
\( \left(-23, 0\right) \), \((14,\pm 74)\)
\([-23:0:1]\), \([14:\pm 74:1]\)
\( \left(-23, 0\right) \), \((14,\pm 74)\)
Invariants
| Conductor: | $N$ | = | \( 1776 \) | = | $2^{4} \cdot 3 \cdot 37$ |
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| Minimal Discriminant: | $\Delta$ | = | $-17272267776$ | = | $-1 \cdot 2^{10} \cdot 3^{2} \cdot 37^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{1994709028}{16867449} \) | = | $-1 \cdot 2^{2} \cdot 3^{-2} \cdot 13^{3} \cdot 37^{-4} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.64704223062772207156547247601$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.069419580161100980384445708128$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9705644773886398$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.155678954286463$ | |||
| 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: | $\Omega$ | ≈ | $1.0543809062050462959225425351$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0543809062050462959225425351 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.054380906 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.054381 \cdot 1.000000 \cdot 16}{4^2} \\ & \approx 1.054380906\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1024 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $37$ | $4$ | $I_{4}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.24.0.11 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1776 = 2^{4} \cdot 3 \cdot 37 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 98 & 1651 \\ 895 & 1126 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 64 & 205 \end{array}\right),\left(\begin{array}{rr} 457 & 1560 \\ 1660 & 493 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 4 & 49 \end{array}\right),\left(\begin{array}{rr} 1297 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 494 & 1763 \end{array}\right),\left(\begin{array}{rr} 1761 & 16 \\ 1760 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[1776])$ is a degree-$11195449344$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1776\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 592 = 2^{4} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 48 = 2^{4} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1776a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 888b4, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.4.1-98568.2-b1 |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{1 + \sqrt{37}})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.122825015296.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.9948826238976.36 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.1359880186540032.6 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 37 |
|---|---|---|---|
| Reduction type | add | nonsplit | split |
| $\lambda$-invariant(s) | - | 0 | 1 |
| $\mu$-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 $0$.