Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-348x+2497\)
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(homogenize, simplify) |
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\(y^2z=x^3-348xz^2+2497z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-348x+2497\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 25\right) \) | $1.0338168530395322339413882219$ | $\infty$ |
| \( \left(26, 105\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([6:25:1]\) | $1.0338168530395322339413882219$ | $\infty$ |
| \([26:105:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6, 25\right) \) | $1.0338168530395322339413882219$ | $\infty$ |
| \( \left(26, 105\right) \) | $0$ | $6$ |
Integral points
\((-16,\pm 63)\), \((-9,\pm 70)\), \((6,\pm 25)\), \((8,\pm 15)\), \( \left(11, 0\right) \), \((12,\pm 7)\), \((26,\pm 105)\), \((386,\pm 7575)\)
\([-16:\pm 63:1]\), \([-9:\pm 70:1]\), \([6:\pm 25:1]\), \([8:\pm 15:1]\), \([11:0:1]\), \([12:\pm 7:1]\), \([26:\pm 105:1]\), \([386:\pm 7575:1]\)
\((-16,\pm 63)\), \((-9,\pm 70)\), \((6,\pm 25)\), \((8,\pm 15)\), \( \left(11, 0\right) \), \((12,\pm 7)\), \((26,\pm 105)\), \((386,\pm 7575)\)
Invariants
| Conductor: | $N$ | = | \( 1260 \) | = | $2^{2} \cdot 3^{2} \cdot 5 \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $3704400$ | = | $2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{10788913152}{8575} \) | = | $2^{14} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{-3} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.19074610089408388065165488009$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31495603145959197866956713629$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0135004521922761$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.086112050031145$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0338168530395322339413882219$ |
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| Real period: | $\Omega$ | ≈ | $2.4708965813577122815504004342$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 3\cdot2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5544545279253686398910944152 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.554454528 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.470897 \cdot 1.033817 \cdot 36}{6^2} \\ & \approx 2.554454528\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 288 |
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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 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 2.3.0.1 | $3$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 409 & 12 \\ 408 & 13 \end{array}\right),\left(\begin{array}{rr} 39 & 178 \\ 266 & 401 \end{array}\right),\left(\begin{array}{rr} 152 & 11 \\ 105 & 388 \end{array}\right),\left(\begin{array}{rr} 337 & 12 \\ 342 & 73 \end{array}\right),\left(\begin{array}{rr} 310 & 3 \\ 213 & 412 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 370 & 411 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$46448640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 | $2$ | \( 21 = 3 \cdot 7 \) |
| $3$ | additive | $6$ | \( 20 = 2^{2} \cdot 5 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1260.d
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{12 +2 \sqrt{15}})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.21870000.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.8.4480842240000.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.8782450790400.9 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.67525382385720000.7 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.4691907907061498279186856033600000000.5 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | add | add | nonsplit | split | ss | ord | ord | ord | ord | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 1 | 2 | 1,1 | 1 | 1 | 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 | 0,0 |
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.