Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-681980x+216938022\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-681980xz^2+216938022z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-10911675x+13873121750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(474, -325\right)\) | \(\left(299, 6150\right)\) |
$\hat{h}(P)$ | ≈ | $1.4671109221671479193057586480$ | $1.8450998659539742468300380083$ |
Torsion generators
\( \left(479, -240\right) \)
Integral points
\( \left(-646, 20010\right) \), \( \left(-646, -19365\right) \), \( \left(299, 6150\right) \), \( \left(299, -6450\right) \), \( \left(474, -150\right) \), \( \left(474, -325\right) \), \( \left(479, -240\right) \), \( \left(488, 165\right) \), \( \left(488, -654\right) \), \( \left(560, 2946\right) \), \( \left(560, -3507\right) \), \( \left(1104, 27885\right) \), \( \left(1104, -28990\right) \), \( \left(1503, 50096\right) \), \( \left(1503, -51600\right) \)
Invariants
Conductor: | \( 74025 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 47$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1032908994140625 $ | = | $3^{8} \cdot 5^{10} \cdot 7^{3} \cdot 47 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{3079572809565169}{90680625} \) | = | $3^{-2} \cdot 5^{-4} \cdot 7^{-3} \cdot 47^{-1} \cdot 73^{3} \cdot 1993^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.9805208277112027533570618494\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.62649572716009772035905956433\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9164038144478317\dots$ | |||
Szpiro ratio: | $4.629962205531929\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.4287051856848184300480612911\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.45850491520207851711799504991\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Tamagawa product: | $ 24 $ = $ 2\cdot2^{2}\cdot3\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 6.6814395912795962009975562933 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 6.681439591 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.458505 \cdot 2.428705 \cdot 24}{2^2} \approx 6.681439591$
Modular invariants
Modular form 74025.2.a.j
For more coefficients, see the Downloads section to the right.
Modular degree: | 663552 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $4$ | $I_{4}^{*}$ | Additive | 1 | 2 | 10 | 4 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$47$ | $1$ | $I_{1}$ | Split 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 |
---|---|---|
$2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 39480 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 47 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 31583 & 13152 \\ 34212 & 13127 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 4939 & 4938 \\ 27978 & 34555 \end{array}\right),\left(\begin{array}{rr} 10924 & 13161 \\ 303 & 26326 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13159 & 0 \\ 0 & 39479 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7528 & 3 \\ 37605 & 13162 \end{array}\right),\left(\begin{array}{rr} 8233 & 34548 \\ 21390 & 27967 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 39474 & 39475 \end{array}\right),\left(\begin{array}{rr} 39473 & 8 \\ 39472 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[39480])$ is a degree-$7095413983150080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/39480\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 74025ba
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 4935e1, 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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{329}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{705}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{105}) \) | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{105}, \sqrt{329})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.13156761200625.12 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ord | add | add | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | 4 | - | - | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |
$\mu$-invariant(s) | 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.