Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+284294583x+10537446745741\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+284294583xz^2+10537446745741z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4548713325x+674401140440750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4385, 3442859)$ | $5.3892309403642069414550069900$ | $\infty$ |
| $(-70669/4, 70669/8)$ | $0$ | $2$ |
Integral points
\( \left(4385, 3442859\right) \), \( \left(4385, -3447244\right) \)
Invariants
| Conductor: | $N$ | = | \( 34650 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-49439539819779220881000000000$ | = | $-1 \cdot 2^{9} \cdot 3^{8} \cdot 5^{9} \cdot 7^{4} \cdot 11^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{223090928422700449019831}{4340371122724101696000} \) | = | $2^{-9} \cdot 3^{-2} \cdot 5^{-3} \cdot 7^{-4} \cdot 11^{-12} \cdot 13^{3} \cdot 83^{3} \cdot 56209^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1862661205959987586204965975$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8322410200448937256224943124$ |
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| $abc$ quality: | $Q$ | ≈ | $1.055037962862197$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.030967102696117$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.3892309403642069414550069900$ |
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| Real period: | $\Omega$ | ≈ | $0.026639745466039320269604179445$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 1\cdot2\cdot2\cdot2\cdot( 2^{2} \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4456257722161489025003053449 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.445625772 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.026640 \cdot 5.389231 \cdot 96}{2^2} \\ & \approx 3.445625772\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 31850496 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
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 | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 5528 & 9219 \\ 285 & 374 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 6934 & 3 \\ 6975 & 34 \end{array}\right),\left(\begin{array}{rr} 1926 & 409 \\ 4235 & 2696 \end{array}\right),\left(\begin{array}{rr} 2521 & 24 \\ 2532 & 289 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 1539 & 3076 \\ 9220 & 4539 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 24 \\ 7932 & 289 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 34650.n
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310.h8, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{5})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-6})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.53166243375.4 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.5308416000000.82 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3317760000.7 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.113680048091082032937459064568027618408203125.1 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | nonsplit | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | - | - | 1 | 2 | 1 | 1 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | - | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.