Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-274800x-55441375\)
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(homogenize, simplify) |
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\(y^2z=x^3-274800xz^2-55441375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-274800x-55441375\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-305, 0)$ | $0$ | $2$ |
Integral points
\( \left(-305, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 17100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $239812751250000$ | = | $2^{4} \cdot 3^{12} \cdot 5^{7} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{12592337649664}{1315845} \) | = | $2^{20} \cdot 3^{-6} \cdot 5^{-1} \cdot 19^{-2} \cdot 229^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7912023144763029675563290677$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.20612815373854949808591607547$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0493645006386834$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.046254081086409$ | |||
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: | $\Omega$ | ≈ | $0.20852018217317530855912725851$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 3\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.5022421860781037027095271021 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.502242186 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.208520 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 2.502242186\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 99 & 478 \\ 686 & 1061 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1130 & 1137 \\ 939 & 8 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1090 & 1131 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 781 & 12 \\ 126 & 73 \end{array}\right),\left(\begin{array}{rr} 1129 & 12 \\ 1128 & 13 \end{array}\right),\left(\begin{array}{rr} 751 & 1138 \\ 702 & 1127 \end{array}\right)$.
The torsion field $K:=\Q(E[1140])$ is a degree-$2836684800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1140\Z)$.
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 | $2$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 17100z
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1140c1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.1039680.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.58644450000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.1871424000000.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.27023362560000.11 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.27023362560000.68 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.472063277904164505486127215000000000000.1 | \(\Z/18\Z\) | 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 | 19 |
|---|---|---|---|---|
| Reduction type | add | add | add | split |
| $\lambda$-invariant(s) | - | - | - | 1 |
| $\mu$-invariant(s) | - | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.