Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-1878x-31492\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-1878xz^2-31492z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-2433915x-1461989034\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(274, 4340)$ | $2.8408009515504792549664045471$ | $\infty$ |
Integral points
\( \left(274, 4340\right) \), \( \left(274, -4614\right) \)
Invariants
Conductor: | $N$ | = | \( 4598 \) | = | $2 \cdot 11^{2} \cdot 19$ |
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Discriminant: | $\Delta$ | = | $-269277272$ | = | $-1 \cdot 2^{3} \cdot 11^{6} \cdot 19 $ |
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j-invariant: | $j$ | = | \( -\frac{413493625}{152} \) | = | $-1 \cdot 2^{-3} \cdot 5^{3} \cdot 19^{-1} \cdot 149^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.58550495868831916158768847214$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.61344267771086611044328331684$ |
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$abc$ quality: | $Q$ | ≈ | $0.9328072208159958$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.058654142698318$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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Mordell-Weil rank: | $r$ | = | $ 1$ |
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8408009515504792549664045471$ |
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Real period: | $\Omega$ | ≈ | $0.36260591232978401916525508687$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 3\cdot2\cdot1 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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Special value: | $ L'(E,1)$ | ≈ | $6.1805473247056805984098307297 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.180547325 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.362606 \cdot 2.840801 \cdot 6}{1^2} \\ & \approx 6.180547325\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2160 |
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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 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
$11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 45144 = 2^{3} \cdot 3^{3} \cdot 11 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 41039 & 0 \\ 0 & 45143 \end{array}\right),\left(\begin{array}{rr} 11287 & 4158 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36059 & 4785 \\ 39259 & 39436 \end{array}\right),\left(\begin{array}{rr} 45091 & 54 \\ 45090 & 55 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 39322 & 38383 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4588 & 41085 \\ 27643 & 1354 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 24652 & 24651 \\ 34353 & 28216 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[45144])$ is a degree-$606596677632000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/45144\Z)$.
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 |
---|---|---|---|
$2$ | split multiplicative | $4$ | \( 2299 = 11^{2} \cdot 19 \) |
$3$ | good | $2$ | \( 2299 = 11^{2} \cdot 19 \) |
$11$ | additive | $62$ | \( 38 = 2 \cdot 19 \) |
$19$ | nonsplit multiplicative | $20$ | \( 242 = 2 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 4598n
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 38a3, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
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$2$ | \(\Q(\sqrt{-11}) \) | \(\Z/3\Z\) | 2.0.11.1-1444.1-b1 |
$3$ | 3.1.152.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.4683345777.1 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.173457251.1 | \(\Z/9\Z\) | not in database |
$6$ | 6.0.30751424.1 | \(\Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$12$ | 12.0.168305397188049.1 | \(\Z/9\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.2.9721108584010478577551457671790723072.2 | \(\Z/6\Z\) | not in database |
$18$ | 18.0.493883482396508590029541110185984.1 | \(\Z/18\Z\) | 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 |
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Reduction type | split | ord | ss | ord | add | ord | ord | nonsplit | ord | ord | ord | ord | ss | ord | ss |
$\lambda$-invariant(s) | 3 | 3 | 3,1 | 1 | - | 1 | 1 | 1 | 3 | 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 | 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.